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French prosecutors say a terror investigation has been opened after a gunman opened fire at Christmas market in Strasbourg, leaving two people dead and up to eight wounded, including several in critical condition.

The Paris prosecutor, who is in charge of anti-terror probes in France, is heading to Strasbourg, according to a statement from his office. The prosecutor's office says the investigation is for murder and attempted murder in relation with a terrorist enterprise.

The gunman, who is at large, has been identified and has a criminal record. The prefect of the Strasbourg region says the gunman had been flagged as a suspected extremist.

Police have yet to release a name for the suspect.

France has been on high alert following a wave of attacks by militants inspired by the Islamic State group and al-Qaeda in recent years.

In 2016, a truck was deliberately driven into a packed Christmas market in Berlin after the assailant Anis Amri claimed allegiance to IS.

In recent months France has also been rocked by mass anti-government protests organised by the 'Gilets Jaunes', or Yellow Vest movement.

Five people have died since the start of the protests, which began over increases in fuel tax but have since snowballed into a nationwide opposition force to Macron's agenda and governing style.

The group Tuesday vowed to press on with their demonstrations, a day after wringing out fresh concessions from President Emmanuel Macron that are set to cost the state an estimated 11 billion euros ($12.5 billion).