This main CBD stretch is often crowded with pedestrians and shoppers, and potentially workers given the time of the day the incident took place.
No injures have been reported.
A 50-year-old woman, the driver of one of the vehicles, had been spoken to by police and was charged with reckless driving, the spokeswoman said.
However police were still appealing for the public's help in locating the other car, and its driver which were involved in the dispute in Napier on Thursday afternoon.
The second vehicle was a 1999 green or blue Honda Accord with the registration plate YS5189.
The Accord had visible panel damage and a missing bumper.