Cumbria Police shared a picture of the car wreck after the crash in Edgehill Road, Carlisle on Sunday morning (January 9).
Officers said the baby was not injured in the crash, but he was taken to Cumberland Infirmary as a precaution.
The driver, who has a provisional licence, was arrested and taken into custody.
Posting a picture of the crash scene, Cumbria Police tweeted: "The consequences of failing to stop for Police and driving dangerously all because he had a provisional licence.
"This driver was arrested for multiple offences after plowing into a lamppost and parked vehicle.
"The most shocking part of this was a baby in the rear passenger seat. Amazingly no injuries, thankfully. Incident occurred in Edgehill Road, Carlisle.
"To clarify for media reports, the baby was in the rear of the offending vehicle, not the parked vehicle. He was taken to the CIC as a precaution."
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