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Crews from Temple, Bedminster, Southmead, Kingswood, Patchway and Avonmouth were involved in tackling the fire.
Avon Fire and Rescue Service told the BBC: "The fire has now been extinguished. Firefighters remain on the scene to ventilate the building and ensure the scene is safe."
A spokesperson said the cause of the fire is thought to be accidental and that a number of residents were unable to return home on Tuesday night due to "significant" smoke and water damage.