The Berlin Christmas market attacker pledged his allegiance to Islamic State leader Abu Bakr al-Baghdadi and called for Islamic State supporters to take revenge on the "crusaders" bombing Muslims, a video posted on Amaq news agency showed.

"My message to crusaders bombing Muslims daily-if God is willing, I swear to God that we will come to slaughter you, you pigs," 24-year-old Tunisian Anis Amri said in the video posted online.

Amri is suspected of driving a truck that smashed through a Berlin market on Monday killing 12 people, and security forces across Europe have been trying to track him down.

The truck mowed through a crowd of people and bulldozed wooden huts selling Christmas gifts and snacks beside a famous church in west Berlin.

Militant group Islamic State acknowledged Amri's death and his suspected role in the German attack - for which it has claimed responsibility - through its Amaq news agency.

"The executor of the Berlin attacks carries out another attack on Italian police in Milan and is killed in a shoot-out," it said.

Italian police shot Amri dead, killing him after he pulled a gun on them during a routine check in the early hours of Friday.

A police chief said his men had no idea they might be dealing with Amri when they approached him at around 3 am outside a station in Sesto San Giovanni, a suburb of the northern city of Milan.