STRASBOURG, France -- French authorities detained at least four people as they hunted Wednesday for a suspected Islamic extremist who sprayed gunfire at one of Europe's most famous Christmas markets in the eastern city of Strasbourg. The attack left at least two people dead and 12 wounded, including seven who were in serious condition.

The government raised the security alert level and sent police reinforcements to Strasbourg, where some 350 security forces were searching for the assailant. Police said the shooter was wounded in a gunfight with soldiers after the Tuesday night attack but escaped, and a top official said he might have escaped to neighboring Germany.

French police union officials confirmed on Wednesday to the Associated Press that the suspect was 29-year-old Cherif Chekatt, a Strasbourg resident and native who has a police record in France and Germany, and who had been flagged for extremism. His name had been widely reported by French media since not long after the attack on Tuesday.

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Paris prosecutor Remi Heitz said Wednesday that witnesses reported hearing the gunman shout "Allahu Akbar," Arabic for "God is great," during the attack. Prosecutors have launched a terrorism investigation into the shooting, but authorities have not yet confirmed a motive.

A man identified as native Strasbourg resident Cherif Chekatt was the subject of a manhunt around the eastern French city on Dec. 12, 2018, a day after he allegedly opened fire on the city's Christmas Market, killing at least two people.

"Considering the target, his way of operating, his profile and the testimonies of those who heard him yell 'Allahu Akbar,' the anti-terrorist police have been called into action," Heitz said.

Strasbourg Mayor Ronald Ries was unambiguous, however, declaring on Wednesday that what happened in his city, "was indisputably a terrorist attack."

Cross-border manhunt for a terror suspect

Senior Interior Ministry official Laurent Nunez said on France-Inter radio that police sought to arrest the suspect 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. There have been reports in French media that other members of his family were believed to have been radicalized.

In Germany, police said they had detained three people in a taxi with French licence plates, after tip-offs from the public following the Strasbourg attack. It was not clear whether any of those detained in Germany were in fact linked to the attack.

A law enforcement source told CBS News on Tuesday that the attacker used a long gun in his siege on the Christmas market. His home was raided Tuesday morning in relation to a previous murder investigation, but officers failed to apprehend him in that raid.

Nunez said eight of the injured were in serious condition, and the city mayor said some had head wounds.

Peter Fritz, an eyewitness, told CBS News partner network BBC News that he was in the area when the shooting broke out. "I saw a person with apparently two shots into the head lying on the bridge here," he said. "We tried to engage in resuscitation activities. After 45 minutes, we stopped the resuscitation effort because a doctor told us over the phone that it is senseless."

Other witnesses described shots and screaming after the gunman opened fire at the Christmas market 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.

France's recent history of Islamic extremist violence

The attack was a new blow to France, after a wave of Islamic extremist attacks in 2015 and 2016, and truck attack in Nice that killed dozens in 2016. It also comes 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."

The assailant got inside a security zone around the venue and opened fire from there, Mayor Roland Ries said on BFM television.

Police were out in force in Strasbourg on Wednesday morning, and the Christmas market was closed. The market in Strasbourg, about 310 miles east of Paris on France's border with Germany, is a long-standing holiday tradition. It was the target of an al Qaeda-linked plot at the turn of the millennium.

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 and to send security reinforcements to Strasbourg, Interior Minister Christophe Castaner said in Strasbourg.

Some 350 officers and two helicopters were searching for the assailant after Tuesday's drama, which involved shooting in multiple neighborhoods of Strasbourg, authorities said. The assailant confronted law enforcement officers twice, exchanging fire, while he "sowed terror," Castaner said.

The shooter was shot and wounded by soldiers guarding the Christmas market, according to Stephane Morisse of police union FGP Police.

"I heard two or three shots at around 7:55 p.m., then I heard screams. I got close to the window. I saw people running. After that I closed the shutters. Then I heard more shots, closer this time," said Yoann Bazard, 27, who lives in central Strasbourg.

"I thought maybe it's firecrackers," he said, speaking by phone. "And then, as it got close, it was really shocking. There were a lot of screams. ... There were police or soldiers shouting 'Get inside!' and 'Put your hands on your head.'"