Krzysztof Jan Klęczar, regional governor of Lesser Poland, said the dead included three Polish women aged 47, 53 and 61, a 32-year-old British man, and a woman of 51 whose nationality has not been confirmed but who is not a Polish citizen.
He said the sixth victim was a Polish man around 50 years old.
The shooting in the city's historic Main Square, popular with foreign tourists, began at roughly 3pm local time on Saturday.
The 24-year-old gunman, who was in military-style clothing and wore a bulletproof vest, shot people at random from a black car.
Video filmed in Krakow shows a male police officer appearing to shoot the attacker six times from close range, before collapsing himself.
Other footage showed blood on the floor and bystanders trying to resuscitate two victims. A third person appeared motionless nearby.
Officers 'out of comas'
Two officers who were seriously injured while stopping the attacker were both put into medically induced comas in hospital.
Interior minister Marcin Kierwiński said in a post on X on Sunday afternoon that the pair, who he named as Jakub Kulik and Artur Morański, "have been brought out of comas".
They were, the minister said, "incredibly brave", telling them: "We're rooting for you! Stay strong and get well soon! Poland needs you!"
Mr Klęczar earlier praised their "swift and self-sacrificing" actions, as, but for them, "the number of casualties could have been significantly higher".
Local police earlier appealed for blood donations for Mr Kulik.
Mr Kierwinski said the attacker held "very radical views" and "was fascinated by far-right ideologies".
Family members killed
The bodies of another two women and a man were found at the alleged attacker's home, about 100 miles (160km) northeast of Krakow.
Polish police confirmed on Sunday morning that they were his parents and 21-year-old sister and had likely died from gunshot wounds.
Briton among the injured
Mr Klęczar said a Briton is one of five foreign nationals among the 16 people wounded and in hospital, the other 11 of whom are Polish.
He said one is in a critical condition and another is in a serious condition, while the condition of the other 14 has stabilised.
The other injured foreigners are from Austria, Germany, Portugal and Russia.
The suspect, who had "several" surgeries overnight, was described as "stable but very critical".
Mr Klęczar said investigations - including of the suspect's home - have revealed that "his father legally owned over a dozen firearms". He himself was also a gun owner.
The suspect is yet to be named but Polish interior minister Marcin Kierwinski said it's believed he was influenced by far-right ideology and has "ultra-nationalistic" views.
"The police prevented a much larger massacre that could have otherwise occurred. They fired shots at the attacker just three minutes after receiving the report," the minister said.
Burnham's tribute
Prime Minister Andy Burnham paid tribute to the victims of the attack.
Mr Burnham said: "I'm shocked and saddened by the appalling shooting in Krakow today.
"My thoughts are with the families of those who have lost their lives, everyone injured and all those affected by this senseless violence. The United Kingdom stands in solidarity with the people of Poland."
The FCDO said it is "supporting the bereaved family, and the family of another victim who has been injured".
It is advising British nationals in Krakow to "follow the advice of local authorities and monitor local media".
Attack 'appalling' - Miliband
In a statement, Foreign Secretary Ed Miliband said: "The UK government deeply mourns the British national and other victims who have died after the appalling attack in Krakow.
"All of my deepest condolences go to the families of the victims."
Defence Secretary Wes Streeting called it an "unspeakable tragedy".
Police said more weapons and ammunition were found in the gunman's car, in Krakow.
Polish Prime Minister Donald Tusk said authorities suspected "a significant amount of weaponry" might be inside the the gunman's home in Ostrowiec Swietokrzysk.
Police spokesperson Andrzej Fijolek told broadcaster TVN 24 the attacker used an AR-15 rifle in the Krakow attack and that he owned a firearms licence.
Poland began two days of national mourning on Sunday.
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Mass shootings are uncommon in Poland and the number of gun licences is low compared with some of its neighbours - around 411,000 among 37 million people.
However, the country did suffer a spate of stabbing attacks in schools in the Mazovia province last month, leaving one dead and many wounded.
Authorities said the incidents were influenced by violent and extremist online content.
One train was traveling between Giza and Fayoum when it crashed into the rear of another train going from Cairo to Aswan, authorities said.
Several carriages derailed after the crash in the al-Ayyat district in Giza province, about 32 miles south of the country's capital, the Transportation Ministry said.
Those injured have been taken to nearby hospitals for treatment, while emergency teams and cranes have been sent to the scene to clear the wreckage and restore train services to their normal schedule as soon as possible, the Egyptian Railway Authority said.
Egypt President Abdel-Fattah el-Sissi has ordered a prompt investigation into the crash to hold those responsible accountable.
In a statement on social media, he also expressed his condolences to the families of those who died and wished those injured a speedy recovery.
Last September, a train driver and his assistant were injured after two freight trains collided in the same district.
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Kyiv's drone killed a 40-year-old man operating a tractor in the Kursk region, regional governor Alexander Khinshtein said on Sunday.
On the same day, a Russian drone struck emergency workers on rescue missions in southern Ukraine, hitting a former school and nearby shelter.
It was the same area - in the village of Komyshuvakha, near the frontline in the Zaporizhzhia region - where at least 55 people died last Monday after multiple Russian guided bombs struck as people waited for humanitarian aid, authorities said.
Ukrainian President Volodymyr Zelenskyy said in a post on X on Sunday that Moscow hit the village with an FPV (first-person view) drone, "where emergency rescue operations have been underway for seven days".
It showed Moscow's "absolute cynicism and brutality", as "unfortunately, more people may still be under the rubble".
A Russian attack on energy infrastructure in Ukraine's Dnipropetrovsk region had left more than 600,000 people without power. Other regions also faced outages, Mr Zelenskyy said.
Russia said in response that the strikes by its forces had hit three cargo vessels that were carrying military goods for Ukraine and two electricity substations, which it said supplied power to the Ukrainian defence industry.
According to its defence ministry, one of the container ships had delivered dual-use cargo to Ukraine. It said a third vessel, a dry cargo ship hit south-west of Odesa, was carrying weapons and military equipment for Ukraine's armed forces.
Reuters, which reported Russia's claims, said it could not independently verify the information..
No one was injured in Sunday's strike on Komyshuvakha, but specialised rescue equipment was damaged, according to Ukraine's State Emergency Service.
Around 50 people were still listed as missing from Monday's attack, Ukraine's military intelligence agency, HUR, said. Many of the bodies recovered have not been identified and DNA analysis is ongoing.
Mykola Krasnov, 72, who lives in the village, said he was still looking for his wife, but had heard no word from rescuers.
"She went to get humanitarian aid, and a three-ton bomb was dropped there. Where is she?" he asked. "She didn't come home, and they didn't find her body there. She isn't in the morgue, and she isn't here alive."
Since Russia's full-scale invasion in 2022, Ukraine has often accused Moscow of launching multiple strikes at the same site in quick succession to target emergency crews responding to the initial attack.
The Russian-controlled Zaporizhzhia nuclear power plant, the largest in Europe, has lost its external electricity supply after a damaged power line was disconnected for safety, the plant's management said on Sunday and was being powered by backup diesel generators.
The plant went offline soon after being captured by Russia early in the war, but needs power to keep essential cooling systems running.
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Meanwhile, Europeans will take part in US-mediated talks on Ukraine in Abu Dhabi next week, France's foreign minister has said.
On Saturday, it was announced that negotiators from Russia, Ukraine and the US would meet on Tuesday in the capital of the United Arab Emirates, for talks aimed at ending the war.
Jean-Noel Barrot told radio station RTL: "The Europeans will be well represented; France will be in Abu Dhabi to take part in these discussions.
"I can't tell you yet who will be representing France on this occasion, but France will certainly be there."
The Department for Education (DfE) has suggested the current curriculum puts too much pressure on children to prepare for their GCSEs early, with a survey by Public First showing more than half of teachers say qualifications dominate the first three years of secondary school.
The new curriculum for pupils aged 11 to 14 will slim down "over-crowded" content and allow space for schoolchildren to develop skills they will need for the future, like confident speaking, reading for pleasure and numeracy, the DfE has said.
It also said there will be a more engaging offer in subjects like drama, PE, music, and modern foreign languages, alongside technical and digital skills.
The DfE believes the changes will help tackle persistent absence, which nearly doubles from Year 7 to Year 9.
Teachers will still be able to stretch high-achieving pupils or groups who are ready to go further without narrowing their options, the government has said.
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Education secretary Lucy Powell said: "The early years of secondary school should be a time when children are excited to go to school and engage in the specialist and much broader teaching on offer, and where they find their passion and develop a love of learning.
"Instead, we've seen worrying trends of disengagement and high levels of persistent absence, with the disadvantage gap stubbornly large and too many leaving school without the knowledge and skills they need to thrive.
"That's what our exciting and engaging new national curriculum aims to address as part of our wider education reforms.
"What's more, the pressures of GCSEs and an overcrowded curriculum are resulting in too many children receiving a narrower offer during Key Stage 3 (Year 7-9) and being rushed towards exams without grasping the fundamentals or developing deeper understanding. I'm putting a stop to that."
The government said the changes, which are due to be published in full later this year, are aimed in particular at improving outcomes for white working class pupils, for whom disengagement in the transition to secondary school is markedly worse, according to an independent inquiry.
A yellow weather warning forecasting fog across large parts of the country is in place from 6pm on Sunday until 11am on Monday, with flight disruption possible.
The warning area, stretching from York to Bath and covering most of the South East and eastern England, will likely see the thickest fog, but low-lying cloud will still affect other areas.
The warning means travellers could face slower journey times, with delays to bus and train services possible, while there is also a chance of delays or cancellations to flights.
Niamh Murray, Met Office operational meteorologist, said: "There is still going to be patchy fog elsewhere, just not quite as dense or as impactful.
"Parts of the West Country could still see fog, Northern Ireland, really anywhere across much of the UK, even into the central belt of Scotland."
"With autumn and winter and getting fog during this time of year, we get much longer nights, and that allows temperatures to sort of fall away," Ms Murray said.
"We get a longer period of cooling and also the temperatures are much lower, so they're much closer to the temperature that we need for the air to become saturated and therefore form fog."
Ms Murray added: "We just have the perfect recipe at the moment: light winds, clear skies, long nights, to allow that fog to develop. And particularly if we see the negative numbers, it can quite stubborn to clear the next day."
Fog may be particularly stubborn overnight into Monday in rural parts of the Midlands, Wales and East Anglia, where temperatures will drop close to freezing, with the potential for air frost.
Visibility could be down to 50m or less in some places, the Met Office warned, with delays to bus and train services possible and potential delays or cancellations to flights.
Sky meteorologist Kirsty McCabe said dense fog is likely to cause issues for many over the next couple of nights "until the winds pick up as low pressure moves up from the southwest, bringing milder and wetter weather".
She said: "Later next week we could well be talking about frost instead of fog with the winds set to swing round to a northerly direction."
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Ms McCabe added: "An area of high pressure sitting over the UK this week has effectively trapped all the mist and murk, making it harder for the fog to clear."
She said the situation hasn't been helped by smoke particles from bonfires and fireworks, which make mist and fog thicker and reduce visibility.
On Saturday, a stretch of the M20 was temporarily shut, and two people were seriously injured, after more than 30 vehicles were involved in a crash, amid "thick fog" in the area, according to National Highways.
Kent Police said the incident happened on the London-bound carriageway near Hollingbourne, Maidstone, at around 6.41pm.
During the foggy period, travellers are advised by the Met Office to check road conditions before leaving, as well as allowing extra time for the journey and changing plans if necessary.
The agency said drivers should also make sure they know how to switch on fog lights, and check they are working before setting off and check if plane, bus, train and ferry services are affected before leaving, if using them.