Three people have been killed, including a young child, and 15 others injured when a car slammed into pedestrians in the centre of Australia's second-largest city Melbourne.

A man deliberately drove his vehicle into people on Friday at the Bourke Street Mall, but the incident was not "terrorism-related", Victoria state police's acting commander Stuart Bateson told reporters.

"I can confirm that we believe this is connected to an earlier stabbing in the southern metro region. We have the offender in custody. There is no further threat to the public at this stage," said Bateson.

Police earlier chased the driver who was wanted over a domestic assault and driving offences.

Officers eventually rammed the car and shot the 26-year-old driver in the arm, before dragging him from the vehicle and arresting him. Police said the man had a history of family violence and was wanted for a stabbing earlier in the day.

A shop video showed several people diving into a convenience store as the car raced along the footpath.

"We witnessed about half-a-dozen people that ricocheted off the car one way or another. I saw one person fly up almost roof level of the car as they got thrown up against one of the retail stores," Sharn Baylis, 46, told Reuters news agency.

"You could hear the gasping and the screaming from people, then you just started hearing the screams and the crying as it sunk in," she said.

Royal Children's Hospital in Melbourne said five patients were brought in after the incident. Of them, a three-month-old is in critical condition and a two-year-old is in serious condition.

Baylis said she rushed across tram tracks with other bystanders and gave cardiopulmonary resuscitation to a badly hurt man who had been run over.

Television footage from the scene showed a damaged red car with the front doors open and the windscreen smashed.

"He wasn't stopping and people walking were trying to get away and he just kept going and collected people on his way," one unnamed witness told Melbourne's Herald Sun newspaper.

Emergency services treated the incident as a "mass casualty event" and hospitals were put on "Code Red" to accept the injured.

Heavily armed officers were deployed on the streets in the heart of the city, which is hosting the Australian Open tennis tournament.

"Emergency services are assisting multiple patients," Victoria state police said.