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The incident happened 4km north of the town about noon yesterday.
Senior Constable Stuart Dowling, of Oamaru, said the 27-year-old's car left State Highway 1 and then crashed into the pole after she become distracted by the animal.
The woman was travelling under 80kmh, having entered the highway from Works Rd about 280m north.
Network Waitaki engineering manager Brad Fleming said the initial crash did not cause a power outage, but an unplanned outage was required locally south from Pukeuri Junction to make repairs.
Oamaru customers were not affected.
The driver of the vehicle said she was embarrassed by the incident and declined to be named.