The mother of a French policeman killed in an attack last week after he swapped himself for a hostage said Monday she was proud of his actions, and did not feel hate towards his killer.

Speaking to RTL radio on Monday, Nicolle Beltrame said she was not surprised by her 45-year-old son Arnaud's selfless act at a supermarket in southwest France on Friday, saying he was a fierce patriot and proud serviceman.

"Arnaud would not have wanted us to give up," she said. "I'm here to honour him and I'm speaking to honour him, so that his act has some purpose, that we're a bit more human, a bit more tolerant."

She added: "We need to realise that we can't accept everything either."

Arnaud was killed on Friday during a shooting spree and hostage-taking around the towns of Carcassonne and Trebes by Radouane Lakdim, a Moroccan-born French national, that left four people dead and three injured.

Beltrame said when she heard that a policeman had been killed on Friday, she knew it was her son.

"I know Arnaud: loyal, altruistic, and since he was small working for other people, committed to the country," she explained.

It also emerged Monday that Lakdim's girlfriend, like the gunman himself, had been on a watchlist of suspected extremists, sources close to the inquiry told AFP.

The 18-year-old woman, who has been in police custody since Friday evening, was under surveillance by intelligence services, one source told AFP.

Arnaud Beltrame's family said he was always putting other people before himself -, LA GAZETTE DE LA MANCHE/AFP

A legal source had told AFP on Sunday that the woman was answering questions during her detention, and had "shown signs of radicalisation".

A 17-year-old man has also been detained for questioning.

Lakdim, 25, a drug dealer with a criminal record, was on France's extremist watchlist but authorities had concluded that he did not pose a threat.

- 'Show photos of heroes, not killers' -

Asked if she felt hatred towards Lakdim, Nicolle Beltrame replied: "Not at all. I feel indifference and the greatest sense of contempt."

She suggested not even talking about him, "not showing his photo. We should show the photos of heroes, not killers and monsters."

Investigators have found notes at Lakdim's home in Carcassonne referring to the Islamic State group, a legal source said, including a handwritten letter in which he claimed allegiance to the jihadists.

Lakdim, armed with a gun, knife and homemade explosive devices, hijacked a car in Carcassonne and shot the two people inside, killing the passenger and leaving the Portuguese driver in a critical condition.

He also shot and wounded a police officer out jogging before heading to Trebes, where he killed the butcher and a customer at the Super U supermarket.

Lieutenant-Colonel Beltrame then offered to take the place of a woman being used as a human shield.

Lakdim was shot dead as police moved in to end the siege, but not before Beltrame had already been shot and stabbed.

He died of his wounds early Saturday, prompting an outpouring of emotion across the country.

President Emmanuel Macron's office has said a national tribute will be organised in the coming days.