Pictures of the aftermath showed wounded victims sprawled across the sand close to the badly damaged vehicle.
The driver has been been arrested but police said there were no indications of a terror attack. The motorist told officers he had an epileptic seizure at the wheel and lost control of his Hyundai.
An eight-month-old girl died and her mother was seriously injured after they were hit by the vehicle.
Witnesses told Brazilian TV channel Globo News that the car raced over a bike path and across the pavement, hitting people and crashing through tables and chairs before coming to a stop on the beach.
The vehicle’s smashed widescreen and dented roof indicated the force with which it hit people.
The popular promenade was crowded with people on a summer night when the crash happened at about 8.30pm local time (10.30pm GMT) on Thursday.
At least four people were badly injured, including two children and an Australian tourist.
The eight-month-old girl was taken to hospital, where she was pronounced dead.
Bystanders reportedly attempted to attack the driver, a 41-year-old teacher from Rio, before police intervened.
One victim, who suffered minor injuries, told O Globo newspaper: “He came at a very high speed before he lost control. Thank God I did not lose my senses. It was a terrible scene. The car caught everyone on the sidewalk.”
Police said they found epilepsy medicine in the driver’s car, adding that he would be tested for alcohol despite saying he had not been drinking.