In the town of Visp, about 30 km (20 miles) down the valley from Zermatt, 20 people were evacuated from their homes after a mud- and snowslide sent debris into a portion of the village, Swiss public television, SRF, reported on Tuesday.
An avalanche risk at its highest level means the stranded tourists will not be able to take advantage of the abundant snow for the time being.
For most of the way the only way in or out was by helicopter airlift, with many tourists choosing to pay 70 Swiss francs (£50) for a seat on a flight.
Zermatt tourism office said teams were working "as a matter of urgency" to reopen roads.
"There are possible power breakdowns all over Zermatt", it said on Twitter, adding that the area was now inaccessible.
Bulldozers have been deployed to help clear the streets of snow.
Up to 1.8m of snow fell inside 36 hours in parts of France's Savoie region, creating avalanches, cutting off towns and causing loss of power.
After the mid-morning trains out of Zermatt was cancelled, visitors began heading towards the resort's heliport, an AFP photographer said.
But the 29 people staying there managed to escape unharmed through the garage.
Switzerland's avalanche research institute said the region has gotten three to four meters (10-13 feet) of snowfall this year. British skier, John Bromell, age 39, from Willingham by Stow near Lincoln, went missing on Sunday at the Tignes ski resort in France Alps. Police about 100 people per hour were being airlifted to safety.