A massive manhunt is under way for Cherif Chekatt after the shooting which left 12 people injured and another brain dead.
The attack began at around 20:00 local time (19:00 GMT), close to Strasbourg's Christmas market, which attracts thousands of visitors at this time of year.
At least three gunshots were fired in the La Meinau district where a police operation was underway on Thursday.
French police say a large-scale manhunt is under way for a suspected extremist gunman.
The anti-terrorist section of the Paris prosecutor's office declared the incident to be an act of terrorism and announced an inquiry had been opened into "murder and attempted murder in relation to a criminal enterprise".
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Cherif C, 29 years old, has been identified as the Strasbourg Christmas Market gunman that left ten people injured.
Castaner also said France had raised its security alert level to "emergency attack" with "the implementation of reinforced border controls and tightened controls on all Christmas markets in France to avoid the risk of a copycat" attack.
The four others detained were Chekatt's parents and two of his brothers.
Officers said he had shot at soldiers patrolling as part of the nationwide Operation Sentinelle, reportedly injuring one of them.
Department of Information chief Busadee Santipitaks said diplomats were helping her with accommodation and will work closely with French police if further developments arise.
A neighbor, who asked not to be named because the gunman was still at large, said he was rarely home.
The man suspected of opening fire at a Christmas market and killing three people has been shot dead, according to reports.
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"[Chekatt] had already been flagged for surveillance by security services for religious radicalization after his time in prison, but [France's] Interior Ministry said he was not suspected of plotting any attacks and they haven't officially called this terrorism yet".
"His family has lived around here for a while, but he lived on his own nearby", a young man named Zach told AFP.
French police officers patrol in Strasbourg, France, on December 13, 2018, following a deadly terror attack in the city two days earlier.
Italian Interior Minister Matteo Salvini said one Italian - Europhonica radio consortium journalist Antonio Megalizzi, 28 - remained in critical condition.
French President Emmanuel Macron was in Brussels on Thursday for a European Union summit.
In November 2015, some 130 people were killed in Paris in coordinated jihadist attacks on the Bataclan concert hall and other sites.
The 64-year-old added: "I found out later that area was used to collect and treat any victims".
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