A massive manhunt involving hundreds of police and soldiers is under way for a suspected extremist who yelled "Allahu Akbar" during a shooting spree around one of Europe's most famous Christmas markets.

The assault in the eastern French city of Strasbourg killed two, left one person brain dead and injured 12 others, authorities say.

Police union officials identified the suspected assailant as Cherif Chekatt, a 29-year-old with a long police record for crimes including armed robbery, and monitored as a suspected religious radical by French intelligence services.

Prosecutor Remy Heitz said the suspected gunman was shot in the arm during an exchange of fire with French soldiers during his rampage in the city centre on Tuesday.

He then took a taxi to another part of the city, boasting of the attack to the driver. There, he exchanged more gunfire with police and disappeared.

Heitz said the man attacked his victims with a handgun and a knife.

Previously, French authorities had said the assailant killed three people, but Heitz said two people were confirmed dead while the third was brain dead. A further 12 people were wou nded, six of them gravely.

Witnesses described shots and screams after the gunman opened fire around the Christmas market on Tuesday evening in a city that's home to the European parliament and considers itself a capital of Europe - and promotes itself as the "capital of Christmas".

For several hours swaths of the city were under lockdown.

Senior Interior Ministry official Laurent Nunez said the suspect had been radicalised in prison and had been monitored by French intelligence services since his release in late 2015 because of his suspected religious extremism.

Nunez said on France-Inter radio that police sought to arrest the man on Tuesday morning, hours before the shooting, in relation to an attempted murder. He was not at home but five other people were detained, authorities said.

Heitz said police seized a grenade, a rifle and knives during the operation.

After the evening attack, as police fanned out in their manhunt, officers also detained four associates of the gunman, the prosecutor said. Witnesses reported that the assailant yelled "Allahu akbar!" (God is great in Arabic) during the attack, he added.

The government raised the security alert level and sent police reinforcements to Strasbourg, where hundreds of police and soldiers were involved in the search.

A terrorism investigation was opened, but the motive of the attack is unclear.

Though authorities urged people in the area to stay inside after Tuesday's attack, Strasbourg mayor Roland Ries told BFM television on Wednesday that "life must go on" so that the city doesn't cede to a "terrorist who is trying to disrupt our way of life".