The incident took place in Common Road on Thursday 17 September 2020 and led to the officer receiving treatment for leg injuries at a local hospital.
At around 1am, a patrol had stopped to speak to two occupants of a grey Ford Kuga, a man and a woman, parked at the roadside.
During a search for suspected drugs, the man is reported to have started manoeuvring the vehicle which then collided with one of the officers before leaving the scene.
Following a search of the area by patrols and a police helicopter, the car was recovered in a layby on Pilgrims Way near the junction with Bull Lane, Eccles. A quantity of suspected drugs was seized from inside the vehicle.
A woman in her 20s from Maidstone was arrested in connection with the incident and taken into police custody.
Enquiries continue to locate the driver and detectives are appealing for anybody with information that may assist the investigation to contact them.
Drivers in and around the area are also asked to check dash cams and residents with CCTV or doorbell cameras are asked to review any footage.
Anybody with information is urged to contact Kent Police on 01622 604100, quoting crime reference 46/166075/20.