Officials offered a reward of up to 100,000 euros ($105,000) for information leading to his arrest. They say he is "under urgent suspicion".
The Islamic State (IS) group has claimed responsibility for Monday's attack outside Berlin's Kaiser Wilhelm Memorial Church in which a truck plowed into a crowd of shoppers, killing 12 people and injuring 56 others. The man was born in 1992, he said.
The White House condemned what it called an apparent terrorist attack.
Germany has launched a Europe-wide manhunt for an "armed and dangerous" Tunisian with ties to Islamic extremists in connection with the Berlin Christmas market attack, according to a European arrest warrant and German lawmakers.
They have arrested the suspected driver and are investigating whether the truck was hijacked from a construction site in neighbouring Poland.
Her comments were echoed by Nigel Farage, the former leader of Ukip. Health officials said 12 of the injured had very serious wounds.
In Washington D.C., State Department spokesman John Kirby said the attack "bears the hallmarks of previous terror attacks", but USA officials didn't have enough information to back up Islamic State's claim of responsibility.
Earlier, German Interior Ministry spokesman Tobias Plate said that Germany was "relieved" by the reports that the suspect in the truck attack had been killed and "doesn't pose a threat anymore".
German Chancellor Angela Merkel thanked Italian forces and said Amri's death would not end the investigation, vowing to pursue "each and every aspect" of the attacker's case. Movio's 29-year-old partner, Luca Scata, fatally shot Amri in the chest. Germany's Federal Criminal Police Office now considers 549 Islamic extremists capable of committing "politically motivated crimes of considerable significance". The group allegedly targeted and radicalized young Muslims in the northwestern states of Lower Saxony and North Rhine-Westphalia.
Police across Europe have been searching for the assailant since the attack on Monday.
A witness, Emma Rushton, said the truck "ploughed through" the market and that people had been crushed.
"We don't know for sure whether it was one or several perpetrators", Germany's federal prosecutor Peter Frank told reporters on Tuesday.
The fingerprints of the dead man match those found in the cab of the lorry, according to reports. The gun used to kill him has not yet been recovered.
Sources said a rail ticket found on the Tunisian's body suggested he had caught a high-speed train from France to the northern Italian city of Turin and then taken a local train to Milan.
The truck's Polish owner, Ariel Zurawski, said his cousin had been driving to Berlin but he could not imagine him being responsible for the crash.
Zurawski identified the slain driver as his cousin.