The shots at the centre and a nearby hairdressing salon shortly before midday on Friday caused panic in the bustling 10th district of the French capital, home to several shops and restaurants and a large Kurdish population. Three others were wounded in the attack that the gunman attributed to his being "racist," the source said.
He was found with a case loaded with a box of at least 25 cartridges and "two or three loaded magazines", the source added. The weapon was a "much-used" US Army Colt 1911 pistol.
On Saturday morning, the Paris prosecutor extended the gunman's period of detention to 24 hours and gave an extra charge of acting with a "racist motive." He is already being held on suspicion of murder, attempted murder, armed violence and violating weapons legislation.
French President Emmanuel Macron said "the Kurds in France have been the target of an odious attack in the heart of Paris" and ordered the Paris police chief to meet with leaders of the Kurdish community.
The gunman, who has a history of racist violence, initially targeted the Kurdish cultural centre before entering a hairdressing salon where he was arrested. Of the three wounded people, one was being given intensive care in the hospital and two were treated for serious injuries.
According to the Kurdish Democratic Council in France (CDK-F), the dead included one woman and two men. Emine Kara was a leader of the Kurdish Women's Movement in France, the organisation's spokesman Agit Polat said. Her claim for political asylum in France had been rejected.
The other victims were Abdulrahman Kizil and Mir Perwer, a political refugee and artist, according to the CDK-F. A police source confirmed that Kara and Kizil were among the victims.