The man held on suspicion of killing three people in Reading is Khairi Saadallah.
The 25-year-old from the town was arrested at the scene and police say they are not looking for anyone else over the terrorist incident.
Security sources told the BBC he is originally from Libya.
Prime Minister Boris Johnson said he was “appalled and sickened” by the attack in a park on Saturday evening.
The prime minister praised the bravery of the unarmed officers who arrested the suspect.
A witness told the BBC he saw a man moving between groups of people in Forbury Gardens, a park in Reading town centre, trying to stab them.
Three other people were injured in the attack at about 19:00 BST on Saturday.
Metropolitan Police Assistant Commissioner Neil Basu, head of counter-terrorism policing, described it as an “atrocity”.
Saturday’s horrifying killings may be another example of what security chiefs call “lone actor” attack where a single individual turns extremist beliefs into murderous actions.
In November last year, the UK’s official threat level from terrorism was reduced from “severe” to “substantial” – meaning it remained likely – but there was no intelligence of an immediate risk to life.
Since then, there have already been three major incidents in which two people have died. Two of those attacks were carried out by lone individuals.
Today, detectives will be interviewing their suspect – and a huge operation will have swung into operation.
Electronic analysts will delve into any social media accounts linked to the suspect; they’ll trawl every call and text message going back years, looking for contacts with extremists.
Intelligence officers at MI5 will review both their open and closed case files on so-called “subjects of Interest”.
A picture will emerge of the suspect’s movements. What led to the attack may be very difficult to identify.
Thames Valley Police said on Sunday morning that the attack was now being treated as terrorism and that counter-terrorism police would be taking over the investigation.
The force’s chief constable, John Campbell, told reporters later that a man was arrested within five minutes of the first emergency call made to police – at 18:56 BST on Saturday.
He said lives had been “devastated” by the incident, but added that there was not believed to be a wider risk to the public and there was nothing to suggest anyone else was involved in the attack.
Of the three injured people, one was seen at the John Radcliffe Hospital in Oxford, where they were discharged without being admitted to hospital.
Two were admitted to the Royal Berkshire Hospital in Reading. One has been discharged, while another remains in a stable condition under observation.
The UK’s terrorism threat level of “substantial” is the third of five ratings at which the threat level can stand.
Source: BBCÂ