Norbert Joos, 55, was guiding a party of five people down the 4,049m mountain in the Spallagrats area at around 2pm on Sunday afternoon when he and two others, who were roped together, fell 160 metres, Graubünden police said in a statement.
Joos was killed in the fall while the other two – a 56-year-old woman and a 58-year-old man, both Italian – were seriously injured and flown to the cantonal hospital in Chur by air rescue service Rega.
Two others in the party, who saw the accident happen, were given psychological care.
The group had stayed in the Tschierva hut the night before and were intending to hike to the Marco e Rosa hut on Sunday.
The exact circumstances of the accident are as yet unclear, but police believe one of the three who were roped together slipped.
Well-known in the area and among the international climbing community, Joos was a hugely experienced mountaineer who ascended the Matterhorn at the age of 12.
Before he was 20 he climbed the three great north faces in the Alps – the Eiger, Matterhorn and Grandes Jorasses.
He went on to summit 13 of the world's 14 peaks over 8,000m without oxygen, only missing Everest, which he attempted in 2008 but failed to reach the summit.
In 2006 he suffered a stroke during the descent from Kangchenjunga on the border between Nepal and India.
After news of his death on the Piz Bernina, fellow alpinists paid respect to Joos on Twitter.
Deeply saddened to read of Norbert's death. I summitted Kanchenjunga with him in 2006 and saw him evacuated... https://t.co/akpQCz55ge— Andrew Lock (@andrewlock_oz) July 12, 2016
Back home, he ran the Norbert Joos Bergsport mountain sports shop in Chur.
Speaking to Blick, Joos' sister said: “The mountains were his life. But risk is always there”.
Over the years he had lost several friends in climbing accidents, including his Swiss compatriot Erhard Loretan, who fell to his death on Swiss peak the Grünhorn in 2011.
Sorry to hear of the death of Norbert Joos, super-strong high-altitude climber who knew when to quit the 8,000ers. https://t.co/G6ZRztEPXG— Ed Douglas (@calmandfearless) July 11, 2016