The driver of the vehicle killed himself, police said, without providing further details. They are not looking for more suspects.
About 30 people are reported injured in the incident, which occurred near the Kiepenkerl statue in the old town. Some are said to be in a serious condition.
Police have closed the area and asked people to avoid the city centre.
They said a suspicious object was found in the van but did not specify what it was. Police spokesman Andreas Bode said it was still too early to declare the incident an attack.
The Kiepenkerl statue is located in a popular pedestrian area in the middle of the historic old town. There are traditional inns and restaurants in the vicinity with outdoor seating.
Photographs posted on social media showed tables and chairs strewn at the scene.
An eyewitness told the BBC that he saw the vehicle drive at speed into a restaurant's seated area, so he was sure it was a deliberate attack.
Ulrike Demmer, the federal government's deputy spokesperson, said the government's thoughts were with the victims and their families.
In December 2016, a lorry ploughed into a crowd at a Christmas market in the German capital, Berlin, killing 12 people.