One person was in a critical condition, according to initial reports from the fire department.
The accident happened in the centre of the Italian city of 1.3 million.
For reasons still unknown, a fully occupied tram on line nine crashed into a building at an intersection.
According to eyewitnesses, the tram was supposed to continue straight ahead but turned and derailed. The driver's cab was severely damaged.
Authorities initially released no further details regarding the identity of the deceased and the injured passengers.
"I thought it was an earthquake," one person who had been on board told the Italian news agency ANSA. "I landed on the floor along with the other passengers. It was terrible."
The public prosecutor's office is now investigating for negligent homicide and bodily harm. The tram, a modern model, is said to have been in service for only a few months.
The tram driver was taken to hospital with injuries. It is speculated that he may have suffered a fainting spell. Before the accident, he is said to have missed a stop. As is customary in such cases, a blood sample was taken for alcohol and drug testing.
The two-week long Winter Olympics took place in Milan, ending last Sunday. Many of the foreign visitors also used the trams.
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