Eight people, including a 12-day-old baby and the driver, were injured during the incident in Marbella.
The Civil Guard said the car sped off when officers tried to stop it at a motorway checkpoint on Sunday evening.
It headed into the Costa del Sol city, where it hit pedestrians on a crowded avenue before smashing into other cars.
A police spokeswoman said the driver and the passenger, aged 27 and 28, held British passports and appeared to be drunk.
Spain’s interior ministry said three people had been injured on the street and four others in a crash as the car tried to get away.
The driver of the car was also seriously injured.
Police said they did not think it was a terror-related incident and both men have been arrested.