Media reports said asylum office papers believed to belong to the man were found in the cab of the 40-tonne lorry used in the attack that killed 12 people.
The man is in his early 20s, known by three different names, and was born in the southern city of Tataouine, the reports said.
He applied for asylum in Germany in April and received a temporary residence permit, according to the daily Sueddeutsche Zeitung.
Twelve people were killed when the Polish-registered articulated truck, laden with steel beams, slammed into the crowded holiday market late Monday, smashing wooden stalls and crushing victims.
The scenes revived nightmarish memories of the July 14 truck assault in the French Riviera city of Nice, where 86 people were killed by a Tunisian Islamist.
Twenty-four people remained in hospital, 14 of whom were seriously injured, Interior Minister Thomas de Maiziere said.
"No one will rest until the perpetrator or perpetrators have been caught," he told ARD public television.
The IS-linked Amaq news agency said "a soldier of the Islamic State" carried out the Berlin carnage "in response to appeals to target citizens of coalition countries".
There was no evidence to back the claim, nor was the perpetrator identified.
Germany is part of a US-led coalition fighting IS in Iraq and Syria.
Tunisia is one of the biggest suppliers of jihadist fighters, with some 5,500 of its nationals believed to be involved in combat in Syria, Iraq and Libya.
In a blow to investigators, federal prosecutors announced today they had to release the only suspect in custody after finding no forensic evidence to link him to Germany's deadliest attack in recent years.
The Pakistani man was detained late Monday after he was reportedly seen jumping out of the truck and fleeing the scene.
But officials had expressed growing doubts over whether they had the right suspect, and he denied the charges under repeated questioning.
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