The man arrested by police following the killing of the MP Sir David Amess has been named as Ali Harbi Ali.
The 25-year-old is being held under the Terrorism Act and officers have until Friday to question him.
Whitehall officials confirmed the man's name to the BBC, and said he was a British man of Somali heritage.
The BBC understands Mr Ali was referred to the counter-terrorist Prevent scheme some years ago, but was never a formal subject of interest to MI5.
It also understands that his father, Harbi Ali Kullane, who was previously an adviser to Somalia's prime minister, has been visited by police who have taken his phone for analysis.
Police officers have spent the weekend searching three addresses in the London area.
It is thought a converted Victorian property in Lady Somerset Road in north-west London is linked to the investigation. Neighbours said officers started searching it late on Friday night.
Further searches, also believed to be part of the inquiry, have been taking place at a property in Bounds Green Road, north London, and another in Cranmer Road, Croydon.
Sir David, who had been a Conservative MP since 1983, was stabbed multiple times during a regular Friday meeting with his Southend West constituents at Belfairs Methodist Church in Leigh-on-Sea, Essex.
Southend councillor John Lamb said he has since spoken to two of Sir David's assistants who were at the constituency surgery with Sir David at the time of the attack.
He described how one was in the room with Sir David taking notes. "All of a sudden there was a scream from her, because the person deliberately whipped out a knife and started stabbing David," he said.
"The other lady who was getting names from people outside, she came running in and saw poor David had been stabbed."
He said both were quite distressed but were "coping quite well" under the circumstances.
Catholic priest Father Jeff Woolnough said he tried to administer last rites to Sir David shortly after the stabbing but police told him he could not enter a crime scene. Instead, he prayed for his friend on the street behind a police cordon.
Ali Harbi Ali was initially arrested on suspicion of murder and held in Essex.
He has since been transferred to a London police station where he was further detained under Section 41 of the Terrorism Act.
Police say they are not looking for anyone else for now.
Early investigations revealed a potential motivation linked to Islamist extremism, police said.
It is thought Ali Harbi Ali did not spend long in the Prevent programme - which aims to stop people becoming radicalised.
Teachers, members of the public, the NHS and others can refer individuals to a local panel of police, social workers and other experts who decide whether and how to intervene in their lives.
Engagement in the scheme is voluntary and it is not a criminal sanction.
Sir David, 69, who was married with four daughters and a son, is the second MP to be killed in recent years following the murder of Labour MP Jo Cox in June 2016.
The latest attack has raised concerns for the safety of MPs, many of whom hold constituency surgeries which anyone can attend.
Home Secretary Priti Patel said MPs had access to a "panoply" of security measures - many of which were put in place after Ms Cox's murder - but said changes could be made to constituency surgeries.
Any measures needed to be proportionate, she told the BBC's Andrew Marr show. "We're here to serve, we're here to be accessible to the British public."
Ms Patel described hearing the news that Sir David had died, saying "our worlds were shattered".
A post-mortem examination of Sir David took place on Saturday, police said.
Meanwhile, Tory MP Andrew Rosindell said the killing of his friend and fellow Essex MP "shouldn't change things in a way that stops us going about our democratic role".
"There's got to be some balance to this. I don't have an answer," he told BBC Breakfast on Sunday. "This is not the Britain I want, this is not the country that we're used to."
Labour's Diane Abbott MP said she would prefer to meet constituents behind a screen to prevent possible stabbing attacks.
Commons Speaker Sir Lindsay Hoyle said he wanted to avoid a knee-jerk reaction but insisted "the best had to come out of this hideous killing".
He said security measures would be reviewed to improve MPs' safety and urged MPs to take up measures already available to them.
Tributes to Sir David have been pouring in from politicians and constituents, with the home secretary saying his "infectious personality" meant he "touched so many lives".
Over the weekend, people have gathered for a candlelit vigil in Leigh-on-Sea to mark Sir David's life and attended a church service to share their memories of him.
Many constituents have reflected on his gentle nature and willingness to listen and to help.
Sir David had long campaigned for Southend to be given city status. On Sunday, Sir Lindsay Hoyle said that would be "a good thing to do" in his memory.