After the crash, the driver, a 35-year-old German man, fled on foot armed with a knife on a busy city centre street but was shot and wounded by police.
Police said a 73-three-year German man who suffered serious injuries died later in hospital.
A statement said a 32-year-old Austrian and a Bosnian woman aged 29 were injured by the car.
"At the current stage of the investigation there is nothing to suggest that it was a terrorist" act, the police said.
Officers had tracked down the driver with help from witnesses and opened fire on the suspect who was "seriously wounded", police said.
They said there was "no additional information on the state of health" of the driver.
German daily Bild reported that the suspect was suffering from psychiatric troubles, but authorities have made no comment on that claim.
The police cordoned off the area and a helicopter hovered overhead on Saturday evening.
Germany has been on high alert since a Tunisian allegedly rammed a truck into a crowded Christmas market in Berlin in December, killing 12 people, before being shot dead days later by police in Italy.
The Berlin carnage evoked memories of the July truck assault in the French Riviera city of Nice, where 86 people were killed by a Tunisian Islamic State group-sympathiser.
According to German security services, there are about 10,000 radical Islamists in the country of whom 1,600 have suspected links to terror groups.