The woman is being treated in hospital for injuries to her leg.
She was hurt as she and another person were walking on the Mount in Shrewsbury at about 11pm on Saturday.
West Mercia police said: "Two people were walking away from the town centre towards the Shelton traffic lights when a small dark car collided with a traffic cone which then struck the victim.
"The woman suffered injuries to her leg and was taken to Royal Shrewsbury Hospital where she is currently in a stable condition."
The driver of the car did not stop at the scene and police say officers are keen to locate them.
Anyone with information or dash-cam footage is urged to call 101 quoting incident 00237i 21092020, go on the West Mercia Police Website or call Crimestoppers anonymously on 0800555111.