Elizabeth McGuinness, 63, from Glen Tower, Motherwell, was walking along a footpath when she was fatally injured at around 10.40am yesterday in the North Lanarkshire town.
Her 64-year-old friend was also hurt and she remains in a "serious" condition at Wishaw General Hospital.
Strathclyde Police said the car, which was going in the wrong direction down a one-way street, struck two other vehicles before mounting the curb in Muir Street and hitting the pedestrians.
The male driver of the Vauxhall Insignia car, who is believed to have lost control of his vehicle after becoming unwell at the wheel, was also treated at Wishaw General Hospital but has since been released.
Sergeant Scott Sutherland, from Motherwell traffic department, said: "It would appear that the driver of the car took unwell at the wheel and, for some reason, drove the wrong way into a one-way street.
"He then collided with a couple of parked cars before mounting the pavement and hitting the two ladies.
"This area of the town was very busy and from CCTV we know that a number of people were in Muir Street when the collision happened and ran to the aid of those involved.
"I would ask anyone who witnessed the crash or who has any information that will help us with our enquiries to contact Motherwell police office on 01698 483000."