One of the heroes of the horrifying Bondi stabbing rampage has been offered a permanent visa to stay in Australia.

Footage of French national Damien Guerot fending off knife-wielding killer Joel Cauchi at Westfield Bondi Junction, armed with just a bollard, went viral in the hours after Saturday’s attacks.

Guerot, who was quickly dubbed “bollard man”, was in the shopping centre on his way to the gym with fellow Frenchman Silas Despreaux.

He has been credited with stopping Cauchi reaching a level that housed a busy children’s playing area.

They heard someone screaming “there’s a man stabbing people”, and ran to see what was happening.

“We just saw him coming … we were thinking, ‘We need to try to stop him’,” Guerot told the Seven Network.

They each grabbed a bollard, and then followed Cauchi down a level.

Guerot said he came face to face with Cauchi on the escalator, and was met with “empty eyes”.

“He wasn’t there,” he said.

“We tried to throw the bollard. We really wanted to stop him.”

On Tuesday, Prime Minister Anthony Albanese thanked Guerot for his bravery.

“I say this to Damien, who is dealing with his visa applications, that you are welcome here … to stay for as long as you like. This is someone who we would welcome becoming an Australian citizen, although that would, of course, be a loss for France,” he said.

“We thank him for his extraordinary bravery. It says a lot about the nature of humanity at a time when we are facing difficult issues, that someone who is not a citizen of this country stood bravely at the top of those escalators and stopped this perpetrator from getting onto another floor and potentially inflicting further carnage on citizens.

“I think that on Saturday we saw some of the best of human character, at the same time as we saw such devastating tragedy. And I thank Damien for his extraordinary efforts.”

Cauchi fatally stabbed five women and a man at Westfield Bondi Junction on Saturday.

The 40-year-old seemingly targeted women, although authorities are yet to pinpoint a motive for the nation’s worst massacre in recent years.

Cauchi was shot dead after killing Dawn Singleton, 25, Jade Young, 47, Pikria Darchia, 55, Yixuan Cheng, 27 and 30-year-old Faraz Tahir, while Ashlee Good, 38, later died in hospital

Also on Tuesday, NSW Police Commissioner Karen Webb said reports Cauchi had visited shopping centres in western Sydney at Parramatta and Penrith in the days before the attack would be looked into.

The investigation was probing his movements in the days and weeks before Saturday, and even examining his childhood to get a better understanding of his thoughts, she said.

“I’ve made it clear we may not ever get an answer but we will get a picture of his movements and what he’s been doing,” Webb said.

As police expand their investigation, a baby girl critically injured in the carnage is showing signs of recovery.

Seven people, including the child, remained in hospital on Monday night.

Health Minister Ryan Park said her condition had improved.

“Pleasingly, we have seen the nine-month-old baby in the children’s hospital at Randwick have her condition downgraded from critical to serious,” he said.

“We hope to have her on a ward over the next few days.”

The Sydney Opera House was lit up on Monday night with a black ribbon in tribute to those affected by the tragedy.

Cauchi’s family say he lived with mental illness, including schizophrenia, for decades.

“He wanted a girlfriend and he has no social skills and he was frustrated out of his brain,” the killer’s father Andrew Cauchi said.

NSW Premier Chris Minns, who has announced a special coronial inquiry to examine the circumstances of Cauchi’s “horrifying, vile act”, also conceded his motive might never be known.

Hundreds of people fled the busy eastern Sydney shopping centre during the terrifying afternoon attack, vision of which was widely circulated on social media.

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-with AAP