North Yorkshire Police said the small blue "old styled" car collided with the 52-year-old-man on Princess Street in Scarborough at around 10.30pm on Sunday (Oct 7).
The collision left the pedestrian with a fractured right ankle and facial injuries.
He was taken to Scarborough District Hospital for treatment.
The blue car did not stop at the scene and was seen to travel from Princess Street to St Sepulchre Street.
Anyone who witnessed the collision, or anyone with information or dash cam footage which would assist police with their enquiries is asked to dial 101 and ask to speak to Traffic Constable Mark Patterson., quoting reference 12180187781.
Crimestoppers can be contacted anonymously on 0800 555 111.