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Sources spoken to by the Otago Daily Times confirmed Sergeant Wayne Taylor had lived in Clyde when he was young and attended Dunstan High School.
Sgt Taylor, who is understood to have been in his 40s, is survived by his wife and their four children.
His wife grew up in Alexandra, where members of her family still live.
It is understood Sgt Taylor still has some family members living in the Central Otago and Queenstown area.
A New Zealand Defence Force statement said Sgt Taylor died during a training exercise.
A call came in from a vessel near Channel Island about 6.30am yesterday and an Auckland Rescue Helicopter was dispatched to the scene shortly afterwards.
Chief of Army Major-general Peter Kelly said Sgt Taylor would be remembered as "an outstanding soldier, leader, father, family man and friend to many, as well as being an all-round top bloke".
"He was a consummate professional, who was known for his dedication and reliability — always upholding our core values in every endeavour.
"I know that his family, friends and workmates will be keenly feeling the loss of a husband, father and friend.
"My thoughts are with them all over the coming days and weeks," Maj-gen Kelly said.
His family were being supported by members of New Zealand Special Operations Force and the New Zealand Army.
Sgt Taylor joined the Royal New Zealand Infantry Regiment as a member of the Army Reserve in 1993 and joined the Regular Force in October 1997, then served in East Timor and Afghanistan.
The special operations force (NZSOF) is "highly capable" and works to pre-empt threats to New Zealand domestically and abroad, according to the defence force. It includes the SAS.
In 2012, Private Michael Victor Ross, of Kaitaia, died during a training exercise at Lake Moawhango.
The Court of Inquiry found while Pte Ross’ death was accidental, a lack of adherence to safety procedures contributed to him falling overboard and drowning.
A police spokeswoman said police were made aware of a death and would be making inquiries on behalf of the coroner.
- Pam Jones and NZME