The Alfred Hospital is treating five patients injured in the incident. Three are critical but stable, two are seriously injured, the hospital tweeted.
Christine Nixon, former police commissioner of Victoria Police, was on the scene a minute after the incident just before 1.40pm. She said a mother repeatedly begged her, "Where is my baby?" She responded, "I don't know."
Ms Nixon also helped direct traffic afterwards.
A Victoria Police spokeswoman told news..com.au: "We are desperately trying to confirm whether those reports are correct that the baby has died and we have not been able to do so as yet."
Melbourne's busy city centre is in lockdown. The driver, who was arrested dressed only in red underpants, reportedly mounted the footpath before mowing down pedestrians as shots were fired, following a stabbing incident last night in Windsor.
Witness Sian Bayliss told ABC News: "One of the first people I rushed over to was a family with two young children who were probably about seven and nine and the little boy had a fairly bad abrasion on his forehead but he wasn't seriously injured, just a lot of blood, but because they had obviously just leapt out of the way or been very close, both children were screaming.
"There was just constant, high-pitched, panicked screaming, the terror in his eyes ... he just looked straight at me while I was talking to his father and just ... screaming. Out of control.
He was looking at me."
She said people were "flying like skittles" as the man "drove right through them" in a "determined fashion."
The driver was arrested at the Elizabeth Street end of the mall after he was dragged from the crashed car by heavily armed police and handcuffed. He had earlier been seen "screaming out the window" and doing burnouts near Flinders Street station.
In extraordinary scenes, people ran for their lives as the car sped towards them. Ambulance and hospital staff were called back into work as the city went on "code red" on Friday afternoon, with residents warned not to approach the CBD.
A Twitter user shared unverified CCTV footage from inside a store of pedestrians fleeing as a car runs them down.
"There was a car that just sped along the footpath and hit people," a witness told police at the scene.
A Victoria Police spokesman warned people to avoid Melbourne's city centre and said this was not a counter-terrorism operation.
"Emergency services are assisting multiple patients but at least one person has been confirmed deceased," the police statement said.
"Pedestrian access has been closed on Bourke Street between William and Swanston streets."
Rishi Hathiramani, a CBD shopkeeper, "There's someone on the middle of the ground and there's blood all over the street.
"I've seen accidents but never this bad. It's pretty brutal."
A number of CBD businesses are in lockdown, while others have been evacuated. "A heap of people ran into my store, it was horrible. Half of my staff are in tears," said a Strandbags manager.
