A 41-year-old Birmingham man was arrested on suspicion of dangerous driving after being stopped by police in Rugby on Saturday.
No one was injured when the car allegedly tried to reverse down Newbold Road but one police car suffered some damage.
The driver was detained and immediately arrested.
A 25-year-old woman from Coventry was also arrested on suspicion of possessing a Class A drug.
A 30-year-old London man, who initially ran away from the scene, was quickly caught by officers and arrested on suspicion of possessing an offensive weapon.
All three have been released on bail until May 19 pending further enquiries.