Emergency services, including 10 fire crews, dashed to the scene in Luton, Bedfordshire, after the blaze broke out at around 4am. Residents were evacuated but one person died from their injuries at the scene.
Further details about the victim, including their gender and age, are yet to be disclosed by police.
Others, including police officers, were taken to hospital, Bedfordshire Live says.
All occupants of the flats have been accounted for, although some others have been taken for treatment in hospital.
Bedfordshire Constabulary tweeted: “We understand this is a difficult time and people will want to know if loved ones are safe.
“We would ask people not to speculate or share images of the incident as it may cause distress to those involved. We will provide regular updates as soon as we have them.”
The fire was extinguished by around 10am.
Bedfordshire Fire and Rescue Service remain at the scene to investigate the cause of the blaze.
Luton Council, as well as the emergency services, say they are “looking after” residents, while the building is made safe.