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Hampshire Police said a man in his 20s was arrested at the scene of the crash on suspicion of causing death by driving without due care and attention.
The double-crewed ambulance belonged to South Central Ambulance Service NHS Foundation Trust (SCAS) and was en route to Southampton General Hospital with the elderly patient.
A male relative of the patient killed when an ambulance struck a tree was treated at scene of the accident and has been conveyed to Southampton General Hospital by ambulance.
A male ambulance crew member has been airlifted to Southampton General Hospital by the Hampshire and Isle of Wight Air Ambulance.
South Central Ambulance Service NHS Foundation Trust said it is conducting appropriate welfare checks with the families of all involved.
A Hampshire police spokesman said: "The road is expected to remain closed for the rest of the day while collision investigators examine the scene and arrange for the vehicle to be recovered."
An ambulance has collided with a tree on the A337 in the New Forest. The incident happened at 9.35am on the Brockenhurst to Lyndhurst road, at its junction with New Park. Four people were on board at the time. Hampshire Police are at the scene. Diversions are in place.
A paramedic and an elderly patient were killed when an ambulance crashed into a tree.
For reasons yet to be established, the ambulance left the road and hit a tree, Hampshire Police said.
The driver of the ambulance - a female paramedic - and an elderly male patient travelling in the rear of the vehicle died at the scene.
A male crew member and a male relative of the patient, who were both in the back of the ambulance, suffered serious injuries.
"Next of kin for those involved are in the process of being informed," a police spokesman said.
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A motorist has been arrested after a paramedic and an elderly patient were killed when an ambulance crashed into a tree in Hampshire.