Police spokesman Peter Nuessmeyer said he could not confirm German media reports that the perpetrator was a middle-aged German man who reportedly had psychological issues.
Another police spokesman, Andreas Bode, told reporters the driver fatally shot himself in the van after the crash. He said the driver’s identity was not yet known and that it was too early to speculate about his motive.
He said police were checking witness reports that other perpetrators might have fled from the van at the scene of the crash.
Police tweeted that residents should “avoid the area near the Kiepenkerl pub” in the city’s historic downtown area where a large-scale police operation was under way.
Mr Bode said six of the 20 injured were in a severe condition.
The city’s university hospital called on citizens to donate blood to help them.
Police also said they found a suspicious object in the van that they were examining to see if it is dangerous.
They told German news agency dpa that was the reason authorities cordoned off such a large area around the crash, which took place at 3.27pm local time on a warm spring day.
The Sueddeutsche Zeitung newspaper also said the suspect’s apartment was being searched for possible explosives.
Munster, a major university city, has about 300,000 residents and an attractive city centre rebuilt after the Second World War.
German news channel n-tv showed a narrow street sealed off with red-and-white police tape.
Dozens of ambulances were near the cordoned-off area. Helicopters were flying overhead.
The Kiepenkerl is not only one of the city’s best-known traditional pubs, but also the emblem of the city, depicting a travelling salesman with a long pipe in his mouth and a big backpack on his back.
German Chancellor Angela Merkel said she was “deeply shocked by the terrible events in Munster”.
In a statement, Mrs Merkel said “everything conceivable is being done to investigate the crime and to support the victims and their relatives. My thanks go to all the responders at the scene”.
Germany’s top security official said he was “dismayed by the terrible incident” and that city and regional police were working hard to work out what happened.
Interior minister Horst Seehofer said federal authorities were in close contact with officials in North Rhine-Westphalia state, where Munster is located.
Lino Baldi, who owns an Italian restaurant near the scene of the crash, told Sky TG24 that the city centre was packed due to a Saturday market and summer-like temperatures, which had risen to 25C (77F) from just 12C (53.6F) a day earlier.
German president Frank-Walter Steinmeier said his thoughts are with the victims and relatives of the crash.
“We have to assume this was a serious act of violence,” he said.
“My deep sympathy goes to all those who have lost a loved one.”