The incident happened at around 11.30pm on Friday 16 October 2020 in Liptraps Lane.
It is alleged there was altercation involving the occupants of two cars after which the driver of one of the vehicles, a red Vauxhall Insignia, collided with a woman standing next to the second vehicle.
Officers also received a report the Vauxhall was involved in a collision with a bus during the incident.
The victim, aged in her 40s, was taken to hospital for treatment to leg injuries.
Detective Inspector Natalie Potter of Maidstone CID said: ‘We are making enquiries to trace the driver of the vehicle that collided with the woman and are appealing for anybody who saw the incident or has information that may assist the investigation to contact us.
‘I would also ask motorists who may have been in the area between 11pm and midnight to check for any dashcam footage that shows a red Vauxhall Insignia.
You can also call the independent charity Crimestoppers in Kent anonymously on 0800 555 111, or complete the online form