The identity of the dead was not immediately known. Officials said a number of detainees escaped during the fire.
A prisoner told the Dutch television station NOS that guards initially did not take prisoners' warnings of a fire seriously and told them nothing was wrong.
"They didn't open the door. They kept us locked up. Our throats started hurting. We were kicking and screaming," said the detainee who was not identified.
Police were searching for an unknown number of escaped prisoners, the report said.
The large prison block is located on the east side of Europe's fourth largest airport and is surrounded by a three-metre fence and barbed wire.
The fire broke out shortly after midnight and raged until 3am, Dutch media reported. Fire fighters and airport police were among the injured, according to the news reports. Four of those injured were hospitalised.
About 350 prisoners were being held in the complex when the fire broke out. Some 43 were in the wing that caught fire. Prisoners were taken to other facilities in nearby cities or moved elsewhere within the complex, the reports said.
Helicopters were called in during the fire to track down possible escaped detainees, Dutch media reported.
Hundreds of cocaine smugglers, mostly from the Caribbean and Curacao, are detained at the airport every year, as are immigrants caught trying to enter the country illegally. Prison cell capacity was greatly expanded in 2002.Reuse content