Youssef Zaghba, a 22-year-old Moroccan Italian, was on Tuesday identified as the third London Bridge attacker.
Like the other two Islamist militants, Khuram Butt, 27, and Rachid Redouane, 30, he lived in the east of the city, where another arrest was made on Tuesday.
Twelve people held earlier have been released without charge.
Police say Zaghba was not a subject of interest in Britain. But Italian media reports say authorities there did inform the UK that he was considered a risk.
Butt, a British national born in Pakistan, was definitely on Britain’s radar and appeared in a British TV documentary broadcast last year called “The Jihadis Next Door”.
UK Foreign Secretary Boris Johnson expressed support for the security services on Tuesday and said more investment was being put into counter-terrorism.
However he acknowledged that questions are being asked about how the attack was allowed to happen.
“People are going to look at the front pages today and they are going to say ‘How on earth could we have let this guy, or possibly more, through the net? What happened? How could he possibly be on a Channel 4 programme, then committing atrocities like this? And that is a question that needs to be answered by M15, by the police as the investigation goes on,” he said.
Saturday night’s rampage, in which three men drove into pedestrians on London Bridge before slitting throats and stabbing people in the bustling Borough Market area, was Britain’s third Islamist attack in as many months.
Butt, Redouane and Zaghba were shot dead at the scene of the attack by officers within eight minutes of police receiving the first emergency call.
At 11 o’clock on Tuesday, a minute of silence was observed in London and nationwide. Among those honoring the seven dead and 48 injured were London medical staff who have lost one of their own.
Australian Kirsty Boden, who worked in a hospital in the British capital, has now been named as one of those killed.
Her family say the 28-year-old nurse died as she ran to help others on London Bridge.