Private Watson-Pickering, 23, from Boosbeck, Redcar, was apparently killed when an armoured vehicle overturned and hit a tree.
He was part of a crew operating an armoured vehicle in a training area near the village of Enford on Salisbury Plain.
Sources say the vehicle overturned and hit a tree during an army exercise. The soldier was was taken to Salisbury Hospital.
The presence of live ammunition on board meant firefighters could not use cutting equipment, so Army engineers rescued those inside, the source added.
It took several hours for the Corps of Royal Electrical and Mechanical Engineers to free the soldiers.
Emergency services had to bring in specialist equipment to right the vehicle and the army had to make it safe before the survivors could be extracted.
Wiltshire Police, the Army and the Health and Safety Executive are investigating the circumstances surrounding the soldier’s death.
A Wiltshire Police spokesperson said: “Our thoughts are with the soldier’s family and friends at this difficult time and we would ask that their privacy is respected.”
A Dorset and Wiltshire Fire and Rescue spokesperson said: “We were called to Enford at 11.57am yesterday morning for a road traffic collision, and crews attended from Pewsey and Ludgershall, together with the heavy rescue unit from Trowbridge.”
The Ministry of Defence said: “It is with sadness that we can confirm the death of a soldier on Salisbury Plain Training Area.
“Wiltshire Police are investigating and therefore it would be inappropriate to comment further. The cause of death will be determined by the coroner.”
Devizes MP Danny Kruger said: “I was very sorry to hear of the death of a soldier on Salisbury Plain yesterday, during training near Enford.
“My sincere condolences to the soldier’s family and to his comrades, and my thanks to the police and fire service for their prompt attendance at the scene.
“While thankfully rare, it is vital that all serious accidents that take place during military training exercises are comprehensively investigated.
“We owe so much to the young men and women who risk their lives for our safety and we must do everything we can to keep them safe as well.”
Last week, Salisbury has been the base for the Army Warfighting Experiment with troops testing out new high technology kit as the British Army adapts to digital warfare which is increasingly becoming more prominent across the world.
Private companies were pitching their latest gear, with soldiers testing out equipment and giving them feedback.