A silver Honda estate failed to stop after the collision on the junction of Oat Street and Cowl Street, Evesham, Worcestershire, at 13:20 GMT on Tuesday.
The officer was taken to hospital with minor leg and head injuries and the abandoned vehicle was found in Longford Close, Bidford-on-Avon, Warwickshire.
A 46-year-old Evesham man was arrested.
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A 49-year-old woman from the Evesham area was later arrested on suspicion of obstructing police, a spokesman said.
A man has been arrested on suspicion of attempted murder after a hit-and-run in which a police officer was injured.