A driver who mowed down a cyclist and a pedestrian outside Melbourne's St Paul's Cathedral on Thursday night refused to stop her car after being confronted by several people.

The driver, a 32-year-old woman of no fixed address, was arrested about four hours after she allegedly hit two people on Flinders Street during peak hour.

Homicide Squad detectives are investigating the circumstances of the fatal shooting in Narre Warren on Wednesday evening. Credit:Paul Harris

She was taken to hospital for a mental health assessment, police said.

A witness, who did not want to be named, said he was cycling down Swanston Street when he saw a white car with no numberplates drive through a red light and hit a 47-year-old South Yarra man who was walking across Flinders Street near the cathedral about 6.30pm.