Police detained the car's driver and later said he had acted alone, with no broader threat seen for the public.
The suspect is a 40-year-old German man from the neighbouring state of Rhineland-Palatinate who did not appear to be politically or religiously motivated, authorities said.
"At this time, we have no indications of an extremist or religious background regarding the specific motivation for the act," Interior Minister of the state of Baden-Wuerttemberg Thomas Strobl told reporters.
"The motivation may rather be rooted in the perpetrator's own personal circumstances. However, this is the subject of ongoing investigations."
German police and prosecutors said on Monday at a joint news conference that the driver intentionally rammed his car into people and is being investigated for murder and attempted murder.
They said the driver shot himself in the mouth when he was arrested and had to undergo medical treatment at a hospital.
He could not yet be questioned.
Broadcaster SWR reported that the suspect had experienced psychological issues.
People were seen lying on the ground at the scene and at least two were being resuscitated, an eyewitness told Reuters.
There did not appear to be a connection to Germany's carnival celebrations, Strobl said.
Those festivities culminated on Rose Monday with a number of parades, although not in Mannheim, which held its main event on Sunday.
Police spokesman Stefan Wilhelm said the suspect drove into people on Paradeplatz, a pedestrian street downtown about noon when workers come for lunch breaks.
Local media reported a carnival market was taking place, meaning more visitors than usual in Mannheim, with a population of 326,000.
Mannheim University Hospital said they were treating three people, two adults and a child, the DPA news agency reported.
It was not immediately clear whether other hospitals received patients.
Images from the scene showed parts of the downtown area cordoned off, with a heavy police presence.
Officers gathered round a badly damaged black car.
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 linked to the Islamic State militant group called for attacks on the events in Cologne and Nuremberg.
German Interior Minister Nancy Faeser cancelled her attendance at the parade in Cologne on Monday, Germany's biggest, due to the events in Mannheim.
"It is a terrible act, as I have said - a horror in broad daylight," Faeser told reporters, speaking alongside Strobl.
Rose Monday, the culmination of the annual carnival season celebrated in Germany's mainly Catholic western and southern regions, features parades of floats that often include comical or satirical references to current affairs.
Dressed in traditional jester costumes and sporting colourful makeup, thousands of partygoers danced through the streets of Cologne, Dusseldorf and other cities in western and southern Germany ahead of the fasting season of Lent.
with DPA and AP
