The federal prosecutor's office said it was offering a reward of up to 100,000 euros ($104,000) for information leading to the capture of the suspect, whom it identified as 24-year-old Anis Amri.
"Beware: He could be violent and armed!" the prosecutor's office said in a statement, in which it described Amri as 1.78 meters (5'8") tall, with black hair and brown eyes.
Amri's father and security sources told Tunisia's Radio Mosaique that he had left Tunisia seven years ago as an illegal immigrant and had spent time in prison in Italy.
In Duesseldorf, Ralf Jaeger, interior minister of the western state of North Rhine-Westphalia (NRW), said the Tunisian appeared to have arrived in Germany in July 2015 and his asylum application had been rejected.
He seemed to have used different names and had been identified by security agencies as being in contact with an Islamist network. He had mainly lived in Berlin since February, but was recently in NRW, Jaeger added.
After being turned down for asylum, the man should have been deported but could not be returned to Tunisia because his documents were missing, he said.
"Tunisia at first denied that this person was its citizen," said Jaeger, adding that German authorities started the process of getting new identity papers in August 2016. "The papers weren't issued for a long time. They arrived today."
The new details added to a growing list of questions about whether security authorities missed opportunities to prevent the attack, in which a 25-tonne truck mowed down a crowd of shoppers and smashed through wooden huts selling gifts, mulled wine and sausages. It was the deadliest attack on German soil since 1980.
Christmas markets have been a known potential target for Islamist militants since at least 2000, when authorities thwarted a plot to attack one in Strasbourg, France. And the modus operandi in Berlin was identical to that of a Bastille Day attack in the French city of Nice in July, when a Tunisian-born man rammed a lorry through a seaside crowd and killed 86 people.
"I AM A BERLINER!"
The market at the scene of Monday's attack, at the foot of the Kaiser Wilhelm memorial church, stayed shut on Wednesday, but some 50 people gathered there with banners that read "I am a Berliner" and sang songs such as "We are the World".
More than 60 other Christmas markets across the German capital re-opened on Wednesday under tightened security.
"We don't want to let the terrorists win. If everyone stays away, they are winning," said tourist Nicki Anning at the Gendarmenmarkt Square in central Berlin.
Berlin authorities said 12 people seriously injured in Monday's attack were still being treated in hospital.
The pre-Christmas carnage at a symbolic Berlin site - under the ruined spire of a church bombed in World War Two - has shocked Germans and prompted security reviews across Europe, already on high alert after attacks this year in Belgium and France.
The possible - though unproven - involvement of a migrant or refugee has revived a bitter debate about security and immigration, with Chancellor Angela Merkel facing calls to clamp down after allowing more than a million newcomers into Germany in the past two years.
Merkel, who will run for a fourth term next year, has said it would be particularly repugnant if a refugee seeking protection in Germany was the perpetrator.
Police initially arrested a Pakistani asylum-seeker near the scene, but released him without charge on Tuesday. It remains unclear whether the real perpetrator was acting alone or with others.
The Polish driver of the hijacked truck was found shot dead in the cabin of the vehicle. Bild newspaper said he had been alive until the attack took place. It also quoted an investigator as saying there must have been a struggle with the attacker, who may have been injured.
ISLAMIC STATE CLAIM
Islamic State has claimed responsibility, as it did for the Nice attack.
The Passauer Neue Presse newspaper 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.
Some politicians have blamed Merkel's open-door migrant policy for making such attacks more likely. The anti-immigrant Alternative for Germany (AfD), which has gained support in the last two years as the chancellor's popularity has waned, said on Tuesday that Germany is no longer safe.
Bavarian Interior Minister Joachim Herrmann told German radio there was a higher risk of Islamist attacks because of the influx of migrants in the past two years, many of whom have fled conflicts in countries such as Syria, Iraq and Afghanistan.
The task of tracking the suspects and the movements of the truck may be complicated by the relative scarcity of security cameras in public places in Germany, compared with countries such as Britain.
The German cabinet on Wednesday approved a draft law to broaden video surveillance in public and commercial areas, a measure agreed by political parties last month after violent attacks and sexual assaults on women.
State surveillance is a sensitive issue in Germany because of extensive snooping by the Stasi secret police in Communist East Germany and by the Gestapo in the Nazi era.
Banner/thumbnail credit: Reuters/BKA