A warrant was issued at midnight. Details were not given but media reports say the suspect is a Tunisian man named only as Anis A, born in 1992.
His residence permit was found in the cab of the lorry.
It has emerged that he was reported to counter-terrorism police last month and had been facing deportation since June.
Reports suggest he have been injured in a struggle with the lorry driver, found murdered in the cab. The attack claimed 12 lives in all.
Police are searching a migrant shelter in the Emmerich area of North Rhine-Westphalia, western Germany, where the suspect's permit was issued.
Chancellor Angela Merkel has met her security cabinet to discuss the investigation into the attack.
The Schengen area covers most EU states plus Iceland, Liechtenstein, Norway and Switzerland.
Anis A is reported to have travelled to Italy in 2012 and then on to Germany in 2015 where he applied for asylum and was granted temporary leave to stay in April of this year.
Ralf Jaeger, the minister of interior of North Rhine-Westphalia, said on Wednesday that the claim for asylum had been rejected in June but the papers necessary for deportation had not been ready.
"Security agencies exchanged their findings and information about this person with the Joint Counter-Terrorism Centre in November 2016," the minister said.
Germany's Spiegel news magazine reports that the suspect was "classified as a so-called danger, a police category of people who are suspected of being capable of an attack, and who were therefore regularly checked".
Tunisia, Mr Jaeger said, had denied Anis A was its citizen, so the authorities had had to wait for temporary passport documentation from Tunisia.
"The papers arrived today from Tunisia," Mr Jaeger added.
At various times he is said to have tried to pass himself off as an Egyptian or a Lebanese, using the names Ahmad Z or Mohammed H (under a German convention, suspects are identified by their first name and initial).
He is said to have been briefly detained in August with fake Italian identity documents.
Sueddeutsche Zeitung reports that the suspect moved within the circle of an Islamist preacher, Ahmad Abdelazziz A, known as Abu Walaa, who was arrested in November.
Some 49 people were also injured when the lorry was driven into crowds at the Breitscheidplatz Christmas market. So-called Islamic State said one of its militants carried out the attack but offered no evidence.
Polish citizen Lukasz Urban was found dead on the passenger seat with gunshot and stab wounds.
Details of the casualties have begun to emerge:
More on the victims
Investigators quoted by German media say there is evidence that, despite being stabbed, Mr Urban wrestled him for the steering wheel.
One official quoted by Bild newspaper said it appeared from the post-mortem examination that the driver had survived up to the attack and was shot dead when the truck came to a halt. No gun has been recovered.
Ariel Zurawski, the owner of the Polish transport company, said he had been asked to identify Mr Urban from photographs.
"His face was swollen and bloodied," he told broadcaster TVN. "It was really clear that he was fighting for his life."
Company manager Lukasz Wasik described Mr Urban as a "good, quiet and honest person" and said he believed he would have defended the lorry "to the end".
Police say they are acting on hundreds of tips from the public and are examining DNA traces from the cab of the truck.
German President Joachim Gauck visited some of the injured on Wednesday.
Speaking outside the Charite Hospital in Berlin, he said: "They should feel that they are not alone and that apart from the doctors here, people across the country are hoping and waiting for them to recover."
Officials released the only detained suspect on Tuesday, saying there was no evidence to link him to the attack.
The IS group claimed the attack through its self-styled news agency, saying it was "in response to calls to target nationals of the coalition countries".
Prosecutor Peter Frank told reporters that the style of attack and the choice of target suggested Islamic extremism.
A manhunt is under way across Europe's Schengen states after prosecutors identified a suspect in the lorry attack on a Berlin Christmas market.