Police identified the driver as a 29-year-old German-Armenian, who eventually crashed the car into a shop window and was detained by bystanders and handed over to the authorities.
Germany's Bild newspaper identified the victim as a teacher who was in the street with a group of her school children.
More than a dozen people were injured, a police spokesperson at the scene in western Berlin said, next to the war-ravaged Kaiser Wilhelm Memorial Church, one of the German capital's best-known landmarks.
Five of the injured were in a life-threatening condition, a fire service spokesperson said.
"A man is believed to have driven into a group of people. It is not yet known whether it was an accident or a deliberate act," police said.
"We are currently on the scene with about 130 emergency personnel," the police added. "The vehicle, a small car, was secured on site."
Investigators were looking into whether the incident was a deliberate attack or possibly an accident with a medical cause, a police spokesperson said, adding that the driver had received some medical treatment.
"It is too early to speculate about the background (of the incident)," an Interior Ministry spokesperson told a regular federal government news conference in Berlin.