Another 13 people were injured in the incident at the Los Llanos base, which sent flames and a plume of black smoke billowing into the air, a Defense Ministry official said.
Most of the victims were not believed to be Spaniards, according to the official, who spoke on condition of anonymity because of ministry rules preventing him from being named.
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Emergency crews were working to douse the blaze and determine how much damage there was to other planes involved in the NATO exercise, the ministry said.
A NATO spokeswoman declined to disclose details, referring questions to Spanish and Greek military officials. The Spanish ministry said the jet that crashed was taking part in a NATO training exercise called the Tactical Leadership Program.
According to a US Air Force Website, TLP was formed in 1978 by NATO's Central Region air forces to advance their tactical capabilities and produce tactics, techniques and procedures that improve multi-national tactical air operations.
The first TLP course was located at Fuerstenfeldbruck Air Base, Germany. It has been based at the Spanish base since June 2009.
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