However, they were looking into tips that the man may have had possible contacts with other extremists, Steltner added.
His office later issued a statement saying the suspect had been moved to the closed psychiatric unit of a jail.
Tuesday's violence comes more than three years after a Tunisian man plowed a stolen truck into a Berlin Christmas market, killing 12 people, an attack that was later claimed by the Islamic State group.
Authorities said the detained suspect allegedly first hit a car then drove into the motorcyclist who received life-threatening injuries.
He allegedly hit a second person on a motor scooter, then used his car to push another motorcyclist into the front of another car, prosecutor Steltner said.
Several German news outlets reported that the man shouted “Allahu akbar,” or “God is great” in Arabic, as he got out of his car.
The Bild daily newspaper reported that he also shouted, “Nobody gets closer, or you will all die.”
The suspect then rolled out a prayer carpet and started praying, daily paper Tagesspiegel reported.
A police officer then approached the man, spoke to him in Arabic, pulled him away from the car and detained him, the paper said.