Authorities in France have launched a terror investigation into the shooting, with interior minister Christophe Castaner saying 350 officers are hunting for the man who was already known to security services.
The 29-year-old alleged gunman, from the city in eastern France, has served prison sentences in France and Germany for common law offences and exchanged fire with law enforcement as he “sowed terror”, Mr Castaner said.
In the wake of the attack, which has also left 11 injured, of which five are in a critical condition, the French government has increased its security alert system Vigipirate to its highest level.
Mr Castaner said: “Three hundred and fifty police and gendarmes are currently on the ground to apprehend the suspect, supported by two helicopters, the RAID (French anti-terror police), the BRI (anti-gang brigade) and the Sentinel force.
“The government has decided to move the security level to “Emergency Attack” with stricter controls at the borders, and the implementation of reinforced controls on all the Christmas markets that are taking place in France to avoid the risk of copycat attacks.”
Authorities have not publicly named the man, but French media have reported he is called Cherif C.
Stephane Morisse, from the FGP Police union, said authorities investigating a robbery went to the alleged gunman’s Strasbourg home earlier on Tuesday, but he was not there.
The suspect was shot and injured by soldiers guarding the Christmas market, but he escaped in a taxi, BFM TV said.
He was said to have entered central Strasbourg via the Corbeau bridge to the south of the city centre, before heading to Rue des Orfevres, a popular shopping street close to the cathedral, where he opened fire at around 8pm local time (7pm GMT).
It is understood the Foreign Office has not yet received any reports of British nationals caught up in the shooting.
In the aftermath, residents were urged to stay indoors and the nearby European Parliament was placed on lockdown.
Antonio Tajani, the president of the parliament, said: “This parliament will not be intimidated by terrorist or criminal attacks. Let us move on. We will continue to work and react strengthened by freedom and democracy against terrorist violence.”
Fans attending a basketball game were also kept behind and began to sing the French national anthem in “homage to the victims”.
The mayor of Strasbourg Roland Ries said the Christmas market will be closed on Wednesday and flags will be lowered to half-mast.
He tweeted: “A book of condolences will be opened at the Town Hall from today.
“I want to thank the people of Strasbourg and visitors to the city for their patience and understanding.”
President Emmanuel Macron, who attended a Interdepartmental Crisis Centre meeting, left without speaking to reporters but later tweeted the “solidarity of the entire nation” is with Strasbourg, the victims and their families”.
France has been hit by several extremist attacks in recent years, including the 2015 Paris shootings which left 130 people dead and wounded hundreds more.
In 2016 a truck was driven into Bastille Day crowds in Nice, killing 86 people and leaving hundreds injured.
Twelve people were killed and 48 were injured when a man drove a truck into crowds at a Christmas market in Berlin in 2016.