Several people were struck by the red Vauxhall Astra in Market Place, Arnold, Nottingham, at 1.12am on Saturday.
Four others have been left with less serious injuries.
The car was abandoned at the scene following the incident, with the driver then fleeing on foot, Nottinghamshire Police said.
Officers have identified a suspect – believed to be a white man – and are in the process of tracking him down, the force added.
The incident is not being treated as terror related and counter terror officers are not investigating, police said.
Detective Chief Inspector Ruby Burrow, of Nottinghamshire Police, said: “There is no indication at this stage of any ongoing threat to the wider public.
“However, we understand the incident will cause concern in the community.
“Officers will remain at the scene today to carry out further inquiries, speak to local people and offer reassurance.
“It is also important anyone with information, or relevant dashcam, CCTV, doorbell or mobile phone footage, gets in touch without delay.”
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Roads were closed in the town centre while police investigate.
Market Place and the junctions from Gedling Road to Derby Street, Hallams Lane and Central Avenue were closed off on Saturday morning.
