The gunman wore full military tactical gear during the attack and was described as having a military-grade weapon – but the government said he was not a member of the Polish armed forces.
More weapons and ammunition were found inside the attacker’s car.
A massive armed police presence was deployed across Krakow on Saturday evening, as security measures were ramped up in the capital Warsaw and elsewhere.
Tusk vowed that any “possible supporters of the perpetrator” would be detained and appealed to the public for information.
Three of the people killed in the square were Polish, one was British and two have not been identified, officials said.
Of the fifteen injured, two were police officers and five were foreign nationals – the interior ministry said they were from Britain, Austria, Russia, Germany and Portugal.
UK Prime Minister Andy Burnham said he was “shocked and saddened by the appalling shooting”.
Witnesses who were close to the attack described panic and confusion around the square as it unfolded.
Cezary, a 24-year-old Polish university student in Krakow, said he did not realise the loud bangs he heard were gunshots until he saw four bodies lying on the street.
“We were still hearing gunshots, a lot of people were screaming,” he said, adding that he saw someone trying to resuscitate a person covered in blood on the ground.
Caitlin Shasby, from Newcastle in the UK, said she heard shooting from her hotel room and was told to take shelter inside as police “barricaded” the street outside.
Another witness told Polish news website Inertia: “I saw shots being fired from the driver’s side window and smoke was coming out.
“I ran to the police station in the main market square – when I burst inside, it was full of people hiding on the floor.”
A period of mourning has been declared in Krakow until 11 November.