The Audi clipped two other vehicles before mounting the pavement, hitting a bollard and then crashing through the window of a Costa coffee shop in Kent, south east England.
A woman in her 70s was declared dead at the scene while five more people were taken to hospital, four with serious injuries and one with minor injuries, officials said.
The driver and a female passenger had to be cut free from the vehicle by fire and rescue workers.
Chief Inspector Roscoe Walford of Kent police said the force was investigating whether the male driver lost control of the car after experiencing a "medical episode".
"We are looking at a number of hypothesis, whether it be the medical condition of the driver - he had a medical episode - or he lost control, or some other reason.
"We are going to forensically examine the scene and look at the route the vehicle's taken - he has crashed into several cars on the way down - so careful examination of that and the entire scene. Witness accounts and statements will be taken from those that saw what happened and eventually the driver of the car will be interviewed to find out exactly what's happened," he said.
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