A man allegedly drove his Toyota into the 33-year-old woman, who was walking towards the car, during the attack on Gamon Street in Seddon about 9.20pm on February 10.
The pair, who did not know each other, had just got into an argument in the drive-thru of a nearby fast food restaurant.
When they left the drive-thru, the woman stopped her car on Gamon Street and got out to approach the man in his Toyota C-HR wagon.
He allegedly then drove forward, hitting her, then fled along Gamon Street.
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The Seddon woman was taken to hospital.
Police are searching for the man, who is perceived to be a Caucasian male with a solid build and a small black beard.
He was wearing a black t-shirt.
Police have released a photo of him and CCTV footage in the hope someone can identify him.