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STRASBOURG, France -- A massive manhunt involving hundreds of police and soldiers was underway Wednesday for a suspected extremist who yelled "God is great!" in Arabic 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 said.

Police union officials identified the suspected assailant as Frenchman Cherif Chekatt, a 29-year-old with a thick police record for crimes including armed robbery and monitored as a suspected religious radical by the French intelligence services. The two officials spoke on condition of anonymity because they weren't authorized to publicly discuss details of the large and ongoing investigation into the attack that set France on edge anew.

The suspect's parents and two brothers, also known for radicalism, have been detained, according to a judicial official.

Reflecting Strasbourg's international nature, the dead included a Thai tourist, and an Italian was reportedly among the wounded. The U.S. government, among others, warned citizens in the area to be vigilant. The city is home to the European Parliament and considers itself a capital of Europe -- and promotes itself as the "capital of Christmas."

Some 720 members of the security forces were hunting for the suspected gunman Wednesday. Prosecutor Remy Heitz said the suspect 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 injured, six of them gravely.

Witnesses described shots and screams after the gunman opened fire around the Christmas market Tuesday evening. They also reported that the assailant yelled "God is great!" in Arabic during the attack, the prosecutor added. For several hours swaths of the city were under lockdown.

Senior Interior Ministry official Laurent Nunez said the suspect had been radicalized 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.

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.

At Chekatt's apartment, in an outer neighbourhood of Strasbourg, the lock of the door was broken at his apartment. Police were guarding the building.

A neighbour, who asked not to be named because the gunman was still at large, said he was rarely home. She said she last saw him Monday from her window, which looks out on a common hallway, and he was with another man.

Young men from the apartment block said they knew him as someone who seemed destabilized by his time in prison. "You can just tell," said one, lightly touching the side of his head. They, too, feared being publicly named because the gunman is still being hunted by police.

The suspected attacker's more than two dozen convictions also included crimes in Germany and Switzerland, according to court documents seen by The Associated Press.

The German government says it has stepped up controls on the country's border with France following the attack, but sees no change to the threat level in Germany.

Strasbourg's Christmas market attracts visitors from around the world. A Thai national, 45-year-old Anupong Suebsamarn, was one of the two killed, according to a Thai Foreign Ministry spokeswoman.

The English-language website of the newspaper Khao Sod said Anupong was the owner of a noodle factory in Chachoengsao province, east of Bangkok, and also sold clothes in the Thai capital's garment district. It quoted his uncle as saying he and his wife had originally planned to visit Paris, but the "yellow vest" protests there prompted them to change plans and go to Strasbourg instead.

Italian media say Antonio Megalizzi, 28, was among the wounded, and is in critical condition. Italian daily La Repubblica reports that he was in Strasbourg to follow the European Parliamentary session.

The attack is a new blow to France, which saw a wave of Islamic extremist killings in 2015 and 2016. It came amid a month of protests against President Emmanuel Macron that have blocked roads around the country, led to rioting in the capital and put heavy strain on police.

While authorities urged people in the area to stay inside after Tuesday's attack, Strasbourg Mayor Roland Ries told BFM television 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."

Many of Europe's deadliest terror attacks in recent years have taken place in France. In response to Tuesday's shooting, the government decided to take the country's attack risk up a level on the official threat index.