The incident happened at around 8pm local time on Thursday evening.
The Twitter account of Brazil’s military police force reports the number of injured and says the driver has been taken into custody.
The car stopped in the sand after it apparently crossed the wide pavement that runs between the beach and a busy road with several people knocked down and lying on the ground.
Officials said the 41-year-old driver lost control of the car, which careered across the crowded promenade, coming to a stop in the sand.
He was arrested by police and taken to hospital. Local media reports the driver may have suffered an epileptic fit.