Both the Berliner Zeitung newspaper and the Berliner Morgenpost reported the truck ran into the market outside the landmark Kaiser Wilhelm Memorial Church tonight.
A photo posted by the Morgenpost showed damaged tables and stalls.
Nine people have been killed and many more injured.
Both newspapers reported that it was not immediately clear whether the incident was an accident or some kind of an attack on the market.
Briton Emma Rushton, who was in the market, saw the lorry rush past her at speed and said it could not have been an accident.
She said she only missed being caught in the chaos because she had climbed up some steps to take a seat.
Ms Rushton said: "The stall that we bought our mulled wine from was completely crushed. People were tearing off wooden panels to get out."
She added: "It was not an accident. It was going 40mph, it was in the middle of the market. There was no way that it could have come off the road and it showed no signs of slowing down."
Police later tweeted to urge people to stay at home and refrain from spreading rumours, adding that officers were working at full speed.
Julian King, European commissioner for the security union, tweeted: "My thoughts are with all those affected and their families in #Berlin tonight."