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Witnesses said people screamed in panic and some were thrown into the air by the car as it crashed through the shopping zone.
"We have arrested one person, one vehicle has been secured," police said, adding that a 51-year-old German suspect from the Trier area had been overpowered within minutes of the incident and was now being questioned.
Prosecutor Peter Fritzen later told a news conference the suspect had drunk a significant amount of alcohol, and authorities were not working on the assumption that there was any Islamist militant motive to the incident.
Trier Mayor Wolfram Leibe said: "It looks as if we are talking about a suspect with mental issues, but we should not pass premature judgement."
The interior minister of Rhineland-Palatinate, Roger Lewentz, said four people had been killed, and four severely hurt people were among the injured. He said the route taken by the driver indicated he had carried out the act on purpose.
Mayor Leibe said a nine-month-old baby was among the dead.
"We have a driver who ran amok in the city," he told public broadcaster SWR earlier.
"I just walked through the city centre and it was just horrible. There is a trainer lying on the ground, and the girl it belongs to is dead," he said. He told broadcaster N-TV that people who saw the incident were "totally traumatised" and the street "looks a bit like after a war".
The incident shocked residents of Germany's oldest town, founded by the Romans more than 2,000 years ago.
The Trierischer Volksfreund quoted a witness as saying a Range Rover was driving at high speed and people had been thrown through the air. It said the car had Trier plates.
It reported that people screamed in panic when the car drove through the street.
Officers were scouring the area in search of evidence, backed by police dressed in flak jackets and carrying sub-machine guns. On the streets, Christmas lights twinkled incongruously.
Germany has tightened security on pedestrian zones across the country since a truck attack on a Berlin Christmas market in 2016 that killed 12 people and injured dozens.
In October 2019, a man opened fire on a synagogue in the city of Halle. After failing to get into the building he went on a rampage outside, killing two people.
In February this year a racist gunman killed nine migrants in Hanau near Frankfurt before killing his mother and himself. Only about a week later, a local man ploughed his car into a carnival parade in the town of Volkmarsen, injuring 61.
Germany has tightened measures to fight the coronavirus, with bars and restaurants closed, but shops and schools are still open.
"What happened in Trier is shocking. Our thoughts are with the relatives of the victims, with the numerous injured and with everyone who is currently on duty to care for the victims," Steffen Seibert, a spokesman for Chancellor Angela Merkel, said on Twitter.