Police said the pursuit was abandoned after less than a minute "due to the speed of the offending vehicle".
The Toyota continued into the central city "at high speed", colliding with a taxi at the Trafalgar St intersection.
"The driver then lost control, narrowly avoiding a number of pedestrians in the area before crashing through the main doors of the ANZ bank," police said.
The 31-year-old male driver of the fleeing car was arrested.
Police said he was a disqualified driver and tested positive for alcohol.
He is due to appear in the Nelson District Court on Monday.
The four occupants in the fleeing vehicle and the driver of the taxi were uninjured.