The incident occurred when a police car tried to stop a Fiat Punto in Dewsbury Road, Beeston, Leeds, shortly after 6.30pm yesterday.
The vehicle made off before losing control and colliding with parked cars and the front of a shop near the junction with Tempest Road.
The car also hit a 34-year-old woman pedestrian who received serious injuries. She was taken to Leeds General Infirmary though her injuries are not believed to be life-threatening.
The driver of the car, a 23-year-old man, was arrested in connection with the incident.
Chief Insp Steve Palmer said: “The area would have been busy at the time and we would like to hear from anyone who witnessed the incident or saw the circumstances leading up to it.
“We have made the Independent Police Complaints Commission aware of the incident and have notified West Yorkshire Police Professional Standards Department, which is normal practice for an incident of this nature.”