Police trying to piece together the motivations of a man arrested over a suspected terrorist attack in Westminster have been given more time to question him.

A search was still under way at a Birmingham address linked to the 29-year-old, understood to be Salih Khater, a British national of Sudanese origin, although searches at two more addresses in the city and another in Nottingham were finished by Wednesday evening.

The suspect was arrested on suspicion of attempted murder on Wednesday, having originally been detained on Tuesday morning on suspicion of preparing an act of terrorism after a silver Ford Fiesta was driven into cyclists at Parliament Square, in central London. He has not been charged.

A warrant of further detention, giving officers until Monday 20 August to continue their inquiries while the suspect remains in custody at a south London police station, was granted on Wednesday by Westminster magistrates court, sources said.

Police said the car travelled to London from Birmingham on Monday night and spent the time between about 1.25am and 5.55am in the Tottenham Court Road area, before moving to Westminster and Whitehall at about 6am. It remained there until the incident an hour and 40 minutes later.

Facebook Twitter Pinterest Metropolitan Police photo of the silver Ford Fiesta. Photograph: Metropolitan Police/PA

In the aftermath of the incident both Cressida Dick, the Metropolitan police commissioner, and Sadiq Khan, the mayor of London, have suggested that Parliament Square could be partly pedestrianised to prevent similar incidents in future.

“You will notice that the security around parliament both in terms of armed officers and police officers and physical barriers has been further enhanced over the last several months and there is more to come on that in further months,” said Dick, speaking before unrelated dawn raids in south-east London.

“Whether that area outside should be pedestrianised further, there should be further physical works done, I think is a matter that will be discussed no doubt between parliamentary authorities, us, the intelligence agencies and indeed the local authorities and the mayor.”

Khan told BBC Radio 4’s Today programme: “I have been an advocate for some time now of part-pedestrianising Parliament Square, but making sure we don’t lose the wonderful thing about our democracy, which is people having access to parliamentarians, people being able to lobby parliament [and] visitors being able to visit parliament.”

Play Video 1:26 How the Houses of Parliament crash unfolded – video report

Kirsty Moseley, 31, one of the closest witnesses to the alleged attack, told the Guardian the driver did not shout or say anything as the car collided with as many as 15 cyclists, and he appeared “focused”.

CCTV footage showed the car swerving across a central reservation near Parliament Square and then accelerating down St Margaret Street before crashing into barriers. Pedestrians were seen leaping out of its path.

Two men and one woman were injured in the crash. One was treated at the scene and two were taken to hospital and later discharged.

The scenes had echoes of the Westminster Bridge attack in March 2017, which prompted the extension of steel and concrete security barriers around the Houses of Parliament. Khalid Masood ploughed a car into crowds on the bridge, killing four people, before stabbing and killing an unarmed police officer, Keith Palmer.