A 20-year-old woman was walking on the cobbles across the Market Place at around 1.30am when a car collided with her at low speed, police said. She was knocked to the ground, but not seriously injured.
The vehicle involved is believed to have been a white, 66-plate BMW. It is believed to have left the Market Place and down King Street, before passing Wetherspoons pub and turning right onto Dundas Street.
Anyone with information is asked to call 101.