Latest: A man suspected of being the gunman who killed three people near a Christmas market in Strasbourg died in a shootout with police on Thursday following a two-day manhunt.
Interior Minister Christophe Castaner said the dead man's identity has not been confirmed yet.
But Mr Castaner said the "individual corresponds to the description of the person sought since Tuesday night", 29-year-old Cherif Chekatt.
A top police official also said "everything indicates" the man was Chekatt.
Mr Castaner said the suspect opened fire on police when officials tried to arrest him.
"The moment they tried to arrest him, he turned around and opened fired. They replied," Mr Castaner said.
A local police official, who also spoke on condition of anonymity, said the man who shot at police was armed with a pistol and a knife.
The shooting occurred in the Neudorf neighbourhood of Strasbourg, where police searched intensively earlier Thursday for Chekatt.
Chekatt is accused of killing three people and wounding 13 on Tuesday night.
Mr Castaner said earlier Thursday that three of the injured had been released from hospital and three others were fighting for their lives.
More than 700 officers were deployed to find Chekatt, who had a long criminal record and had been flagged for extremism, government spokesman, Benjamin Griveaux told CNews television.
Asked about the instructions they received, Mr Griveaux said the focus was catching Chekatt "as soon as possible", dead or alive, and to "put an end to the manhunt".
Security forces, including the elite Raid squad, spent two hours searching in Neudorf on Thursday based on "supposition only" that Chekatt could have been hiding in a building nearby two days after the attack, a French police official said.
Chekatt grew up in Neudorf.
Chekatt allegedly shouted "God is great!" in Arabic and sprayed gunfire from a security zone near the Christmas market Tuesday evening.
Authorities said he was wounded during an exchange of fire with security forces and a taxi driver dropped him off in Neudorf after he escaped.
Prosecutors opened a terror investigation.
So far, five people have been arrested and remanded in custody in connection with the investigation, including Chekatt's parents and two of his brothers.
The Paris prosecutor's office said the fifth, who was arrested Thursday at an undisclosed location, was a member of Chekatt's "entourage" but not a family member.
Police distributed a photo of Chekatt, with the warning: "Individual dangerous, above all do not intervene."
France has raised its three-stage threat index to the highest level since the attack and deployed 1,800 additional soldiers across the country to help patrol streets and secure crowded events.
French authorities said Chekatt, born in Strasbourg, appeared on a watch list of people flagged for extremist views.
They said he had 27 criminal convictions, receiving the first at the age of 13.
The people who died in the attack included a Thai tourist, 45-year-old Anupong Suebsamarn, according to the Thai Foreign Ministry.
Five of the wounded were in serious condition, the prefecture of the Strasbourg region said.
French President Emmanuel Macron was in Brussels on Thursday for a European Union summit.
EU leaders held a minute of silence for the latest victims of a mass shooting in France.
Hundreds of people gathered in Strasbourg's 500-year-old cathedral Thursday evening to mourn and seek comfort.
"Evil does not prevail," Archbishop Luc Ravel said.
"And the message of Christmas has not been contradicted but rather confirmed by Tuesday's dramatic night: Evil and good are both there, but in the end the good will have last word."
Strasbourg's usually busy streets were eerily empty on Thursday morning, with a heavy police and military presence.
Some lit candles and brought flowers to a makeshift memorial at the site of the attack.
"You can feel a very heavy atmosphere due all these events," said resident Lucille Romance.
"People are in a state of shock and are avoiding getting out of their house."
The Islamic State's Amaq news agency claimed the gunman was a "soldier" of the group.
Update 8:44pm: A suspect has been killed in a Strasbourg shootout with French police searching for the alleged gunman who killed three people near a Christmas market on Tuesday.
The shooting occurred in the Neudorf neighbourhood, where police conducted a search earlier for Cherif Chekatt, a 29-year-old born in Strasbourg who police had named as the suspect.
A man opened fire on police on Thursday night, and police responded, killing him, sources said.
Reuters news agency reports the suspected gunman who killed three people at a Christmas market in Strasbourg on Tuesday has been killed according to two police sources— Sky News Breaking (@SkyNewsBreak) December 13, 2018
A local police official said the man who opened fire was armed with a pistol and a knife.
Update 3.55pm: Dozens of officers have sealed off part of southern Strasbourg as they look for a suspected terrorist accused of shooting dead three people.
It is understood the neighbourhood is the last place 29-year-old Cherif Chekatt was seen.
One French police official said security forces, including the elite Raid squad, were taking action based on a "supposition only" that Chekkat could be hiding in a nearby building.
Asked about instructions given to police forces searching for Chekatt, government spokesman Benjamin Griveaux said the focus was on catching the suspect "as soon as possible," no matter whether he was dead or alive, and "put an end to the manhunt".
The number of people killed in a shooting attack on Strasbourg's Christmas market has risen to three.
More than 700 police officers are still searching for the suspect.
French security forces have said they are trying to catch the gunman suspected of attacking a Christmas market dead or alive.
The attack in Strasbourg left three people dead and wounded 13 others, including five who are in a serious condition.
More than 700 officers are involved in the manhunt for 29-year-old Cherif Chekatt, who had been flagged for extremism, government spokesman Benjamin Griveaux said. Prosecutors have opened a terror investigation into Tuesday's attack.
Police have distributed a photo of Chekatt, who was wounded in an exchange of fire with security forces, with the warning: "Individual dangerous, above all do not intervene."
The government raised the terror alert level nationwide and deployed 1,800 additional soldiers across France to help patrol streets and secure crowded events.
Mr Griveaux also called on the "yellow vest" protesters not to take to the streets, as some members of the movement have planned a fifth round of demonstrations on Saturday across France to demand tax relief.
Strasbourg was in mourning, with candles lit and flowers left at the site of the attack. The Christmas market was closed at least again on Thursday.
On Wednesday evening, people prayed and sang in the nearby Protestant Church Saint-Pierre-le-Jeune.
Pastor Philippe Eber said this is a moment "to think of those who died in this city because of violence. We also are thinking about all of those who weep for them, the families".
Strasbourg resident Tassia Konstantinidis said "it's important to have a period of mourning and to remember the victims".
The dead included Thai tourist Anupong Suebsamarn, who was 45.
Italian Interior Minister Matteo Salvini said one Italian was among the wounded, in critical condition. The Europhonica radio consortium said Antonio Megalizzi, 28, was in Strasbourg to follow the session of the European Parliament when he was shot.
French authorities said that the suspect, born in Strasbourg, had run-ins with police from the age of 10 and his first conviction was at 13.
Chekatt had been convicted 27 times, mostly in France but also in Switzerland and Germany, for crimes including armed robbery. He had been flagged for extremism and was on a watch list.