Private Jethro Watson-Pickering, 23, of the 1st Yorks Regiment, was part of a crew operating an armoured vehicle in a training area near the village of Enford, Wiltshire, when the incident happened on Friday.
Wiltshire Police said a joint investigation with the Health and Safety Executive and the Army has been launched.
In a statement, the regiment said: “The Yorkshire Regiment very much regrets to announce the death on Friday October 15 of Pte Jethro Watson-Pickering.
“He was from Boosbeck, Redcar and Cleveland, and was deployed on exercise on Salisbury Plain.
“The thoughts and prayers of the regimental family are with his family.”
A Wiltshire Police spokesman said: “We are investigating the circumstances following the death of a 23-year-old soldier at midday on October 15.
“He was part of a crew operating an armoured vehicle and was taking part in a military exercise on Salisbury Plain Training Area near the village of Enford.
“This is now a joint investigation between Wiltshire Police, the Army and the Health and Safety Executive.
“Our thoughts are with the soldier’s family and friends at this difficult time, and we would ask that their privacy is respected.”
An Army spokesman added: “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.”