Firefighters who responded to the call shortly before 10.00pm local time saw heavy smoke and flames at the front of the building with occupants trapped inside, the fire department said.
Around 70 people resided at the Gabriel House, and some 50 firefighters were mobilised to fight the blaze.
The origin and cause of the fire are being investigated.
Massachusetts Governor Maura Healey expressed her condolences to the victims’ families over “the most horrific news imaginable”.
“I know the people of Fall River are strong and resilient, and now is the time for us to all come together to support one another through this terrible tragedy,” she said on X.
– Agence France-Presse