The mountaineers successfully reached the 4,010-meter (13,155-foot) high Lagginhorn summit, but then plunged several hundred meters (yards) to their deaths after beginning to descend, police in the southern Valais region said.
A sixth member of the group, who had stopped before reaching the summit because he didn't feel well, immediately alerted rescue authorities, but the five climbers died on the spot.
Details on the victims' identities and nationalities were not immediately available. Police spokesman Jean-Marie Bornet said authorities were still confirming IDs and must first notify the next of kin. Information on the nationalities likely won't be released before Wednesday, he added.
Police also had no immediate information on the cause of the accident, but Bornet said local prosecutors are investigating.
The group's sixth member was flown to safety by helicopter, he added.
The Lagginghorn summit in the Saas valley about 10 kilometers (6.2 miles) from the Italian border is said to be one of the less technically difficult peaks in the Alps above 4,000 meters, but snow can make the climb tricky.
Police had no immediate information on the weather conditions at the mountain on Tuesday. Also, it was unclear if the group was on a single rope, which could explain why all perished in the same incident, with maybe one or two people falling and the others being pulled down.