German police are looking for a Tunisian man after finding an identity document under the driver's seat of the truck that ploughed into a Berlin Christmas market, killing 12 people, on Monday evening, news website Spiegel Online reported.
The document was in the name of Anis A., born in the southern city of Tataouine in 1992. The man is also believed to use false names, it said.
Daily newspaper Bild reported he was known to police as a possibly dangerous individual, and part of a large Islamist network.
Police initially arrested a Pakistani asylum-seeker near the scene, but released him without charge on Tuesday. Authorities have warned that the attacker is on the run and may be armed. It is not clear if the perpetrator was acting alone or with others.
The 25-tonne truck, belonging to a Polish freight company, smashed into wooden huts serving mulled wine and sausages, injuring about 45 people.
The Polish driver of the truck was found shot dead in the cabin of the vehicle. Bild reported that he was alive until the attack took place.
It quoted an investigator as saying there must have been a struggle with the attacker, who may have been injured.
Islamic State has claimed responsibility, as it did for a similar attack in July when a Tunisian-born man rammed a truck through a crowd celebrating Bastille Day in the French city of Nice. Eighty-six people were killed, and the driver was shot dead by police.
The head of the Association of German Criminal Detectives, Andre Schulz, told German television late on Tuesday that police hoped to make another arrest soon.
"I am relatively confident that we will perhaps tomorrow or in the near future be able to present a new suspect," he said.
Wednesday's Passauer Neue Presse quoted the head of the group of interior ministers from Germany's 16 federal states, Klaus Bouillon, as saying tougher security measures were needed.
"We want to raise the police presence and strengthen the protection of Christmas markets. We will have more patrols. Officers will have machine guns. We want to make access to markets more difficult, with vehicles parked across them," Bouillon told the paper.
German Chancellor Angela Merkel, who will run for a fourth term next year, has said it would be particularly repulsive if a refugee, seeking protection in Germany, was the perpetrator.
Some politicians have blamed her open-door migrant policy for making such attacks more likely.
The anti-immigrant Alternative for Germany (AfD), which has won support in the last two years as Merkel's own popularity has waned, said on Tuesday that Germany is no longer safe.
Some politicians have also called for changes to Merkel's immigration and security policies after she allowed more than a million migrants to enter Germany in the last two years, many fleeing countries such as Syria, Iraq and Afghanistan.
Bavarian Interior Minister Joachim Herrmann told German radio on Wednesday that there was a higher risk of Islamist attacks because of the influx of migrants.