Three appliances and crews from West Ashland, two from Broughton, one from Newport Pagnell, one from Buckingham and five officers attended the scene at Meads Close, New Bradwell.
Firefighters used two main jets, two hose reels, four sets of breathing apparatus and a turntable ladder in an effort to put out the blaze. One flat was severely damaged by fire and smoke and the remainder of the building was damaged by smoke.
A man suffering from burns to his hands and face was left in the care of South Central Ambulance Service. No-body was seriously injured.
An investigation is to be held into the cause of the fire.