The HP-SDRF, which describes the group as a specialised state force dedicated to disaster response, said it was alerted to the incident on Sunday at around 6pm local time.
The injured man was then transported by stretcher but progress was "extremely slow" and a second and third team had to be deployed because of "increasing fatigue and the challenging terrain".
A spokesperson from the UK's Foreign Office said: "We are supporting the family of a British man who has died in India and are in contact with the local authorities.".
A rescue team of 10 personnel was deployed in what was described as "extremely difficult terrain" locating the victims at 10.30pm with one of the men "in a critical condition".
The rescue team added that "unfortunately, upon arrival at Zonal Hospital, Dharamshala... [the man] was declared dead".
