The pilot Maurice Hammond, 60, is in a serious but stable condition in the Norfolk and Norwich Hospital with his family at his bedside.
A spokesman said they were shocked and offered their sincere condolences to the family of the man in his 80s who died.
Emergency services were called to the field at 3.38pm on Sunday following reports of a crash landing.
It is understood the plane was on fire when firefighters arrived.
They rescued Mr Hammond, the pilot, but the passenger was dead at the scene.
Mr Hammond, who owns a fleet of vintage planes known as the Hardwick Warbirds, was flying his P-51D Mustang – a Second World War plane used by the American air force.
The spokesman for the family said: “They are in distress and they are so distraught for the passenger who lost his life. We extend our condolences to his family.
“Our focus right now is on the terrible tragedy that has occurred to the passenger and his family, and on helping Maurice fight his way back.
“The aircraft was operating normally and Mr Hammond is one of the most experienced pilots of that type of plane in the UK. This type of aircraft is maintained on the level of a Formula One car.”
The spokesman said the family would co-operate fully with the Air Accident Investigation Branch, which will investigate the cause of the accident.
Five fire crews and specialist units were sent to the scene and a blanket of foam was sprayed to stop the flames.
East of England Ambulance Service crews and officers and Norfolk Police joined in the operation.
Area manager for Norfolk Fire Service Garry Collins said the plane, which is though to have been coming in to land, was ablaze when the first firefighters arrived.
He said it was “tragic accident near the end of the runway”.
A witness who was walking his dogs when he saw the plane come down said he heard a small explosion after it disappeared behind some trees.
Charles Christian, who lives close to the airfield, said: “The plane in question was flying in and coming into land quite normally and flew right over my head.
“It flew on toward the runway, which is about a quarter of a mile away, and disappeared behind the trees to land. A few seconds later there was a bang, which I am guessing was the explosion.
“There was no sign of anything happening, there wasn’t a column of smoke or anything. But when it came into land it was working totally fine.”
An spokesman for the East of England Ambulance said: “Our thoughts are with the family and friends involved at this sad time.”
The Mustang aeroplane which Maurice Hammond was flying was named Janie and is among a fleet of vintage planes he owned.
It was one of a batch of ten planes made by North American Aviation in Dallas, Texas, in 1945.
It was shipped to New Zealand in August that year and based at Hobsonville, in Auckland, where it was used by the New Zealand airforce until 1955.
Mr Hammond bought the plane in 1997 and brought it back to the UK for a complete restoration, which took four and a half years. Its first flight in civilian operation took place in July 2001.