The official said it was unclear whether any other suspects were involved.
The police is appealing to the public in the western German city to stay at home.
The incident occurred as crowds gathered in cities across regions including Germany's Rhineland for parades to mark the Carnival season.
A black SUV drove at high speed into people, travelling from the centrally located Paradeplatz square towards the city's landmark water tower, the Mannheim24 news website reported.
A reporter from the DPA news agency at the scene witnessed a person being covered by a sheet among debris.
Police spokesman Stefan Wilhelm said that a driver drove into a group of people in Paradeplatz, a pedestrianised street in Mannheim.
He said "several" people were injured but police cannot yet specify how many were hurt and how badly they were hurt.
"We can confirm that one perpetrator was arrested," he said.
"We can't yet give information on whether there were further perpetrators."
Paradeplatz, a major square in the downtown area, lies at the end of a pedestrianised street in Mannheim, which has a population of 326,000 and lies 85km south of Frankfurt.
Mannheim University Hospital said it has prepared everything for a possible mass casualty incident.
The hospital has implemented its disaster and emergency plan to prepare for the care of the injured.
The police operation followed a "dangerous" incident, a police spokesperson earlier told the broadcaster NTV.
Security has been a key concern in Germany following a string of violent attacks in recent weeks, including deadly car rammings in Magdeburg in December and in Munich last month as well as a stabbing in Mannheim in May 2024.
Police were on high alert for this year's carnival parades after social media accounts connected to the Islamic State militant group called for attacks on the events in Cologne and Nuremberg.
with DPA and AP
