The fire in Barn End Lane, Wilmington, was reported at about 07:40 BST, Kent Fire and Rescue Service said.
At its peak, eight fire crews tackled the blaze in one of the flats, which spread smoke to neighbouring flats. The cause of the fire is not yet known.
The man was pronounced dead at the scene. Eight people were taken to hospital after the fire, South East Coast Ambulance (Secamb) said.
Secamb earlier said nine people were taken to hospital but has since revised its figure.
The Kent, Surrey and Sussex Air Ambulance and nine ambulances were called to the scene.
For a time, Darent Valley Hospital in Dartford asked patients not to attend its accident and emergency department, saying the hospital was dealing with a "major incident".
The Darent Valley later tweeted: "The hospital has been stood down and normal service has resumed in the A&E Department."
A man in his 70s has died in a fire at a flat in a sheltered housing block.