The tram driver from Beckenham has been arrested on suspicion of manslaughter.
British Transport Police said they were investigating whether he fell asleep.
The Rail Accident Investigation Branch (RAIB) said the tram was travelling at a "significantly higher speed than is permitted".
Deputy Chief Constable Adrian Hanstock said: "After liaison with the coroner, we can confirm that at least seven people have lost their lives as a result of this incident."
Emergency services rushed to the scene near the Sandilands tram stop on Addiscombe Road in the small town to free people trapped in the two-carriage vehicle.
Transport for London (TfL) said the incident occurred inside a tunnel.
"The Rail Accident Investigation Bureau is now investigating this incident and will work out how this tragic incident has occurred," TfL said.
Witnesses reported hearing a bang at the scene, where police were called at 6.13am GMT on Wednesday.
The tram appeared to be on its side next to an underpass, and there was a large police cordon in the area with scores of emergency vehicles nearby.
British Prime Minister Theresa May offered her thoughts and prayers and said the government was "closely in touch with the emergency services and other authorities to ensure that all is being done that is necessary to deal with this terrible incident".
London mayor Sadiq Khan said: "My thoughts and prayers are with the friends and families of those who have lost their lives today, and everyone involved in this incident.
"I am in close contact with the emergency services and Transport for London, who are working extremely hard under very difficult circumstances to get the situation under control and treat those who have been injured.
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