Police said the fire broke out at 1.30am as revellers were celebrating in a bar called Le Constellation in the resort in southwestern Switzerland, which locals said was popular with teenagers.
Swiss President Guy Parmelin described the disaster as "one of the worst tragedies our country has ever known" and said most of the dead were young people.
The cause of the blaze, which was initially reported as an explosion, remained unclear but authorities said the fire appeared to be an accident rather than an attack.
Video footage verified by Reuters showed fire spreading from the building, and witnesses described scenes of panic and confusion as people rushed to get out.
The head of the local canton's police, Frederic Gisler, said about 40 people were presumed to be dead and 115 were injured, most of them seriously.
Gisler said during a news conference that work is underway to identify the victims and inform their families, adding that the community is "devastated".
He said it was too early to give details on the identity of the dead and injured but Italian authorities said 16 Italians were missing and about a dozen in hospital.
Two young French women who said they were in the bar told France's BFM TV that they saw the fire start in the basement section of the club after a bottle containing "birthday candles" was held up too close to the wooden ceiling.
"The fire spread across the ceiling super quickly," one of the two women, who identified themselves as Emma and Albane, told BFM TV.
The pair said they were able to climb a narrow staircase to the ground floor and escape the building.
Minutes later, the fire had reached the ground floor too, they said.
BFM showed video of a waitress carrying a champagne bottle with a lit "fountain candle" through the bar, one of many in Crans-Montana, a fashionable ski centre with an array of boutiques, luxury hotels and restaurants.
But the footage did not show the fire breaking out.
Local prosecutor Beatrice Pilloud said an investigation had been opened into the blaze at the bar, which Swiss company records showed was owned by a French couple, but she said it was too early to comment on any possible safety failures.
"There are still many circumstances to be clarified ... The most likely scenario at present is that a widespread fire caused an explosion," she told a press conference.
Witnesses said many of those celebrating in the bar appeared to be from different countries but officials said the grim task of identifying badly burned bodies would take time.
Italy's ambassador to Switzerland, Gian Lorenzo Cornado, told Sky TG24 local authorities had told him the fire was started by someone letting off a firework inside the bar.
On Thursday morning, footage from the street outside showed the area cordoned off, with forensic tents behind white screens set up in front of the bar.
The quiet daylight scene, in which small groups of people huddled together, some in tears, some carrying flowers, was a stark contrast to the panic and confusion that confronted first responders hours earlier.
"The first responders - the firefighters and police officers - arrived at a scene of chaos, at a dramatic scene," Stephane Ganzer, head of security for Valais canton, told reporters.
"What was meant to be a moment of joy turned, on the first day of the year in Crans-Montana, into mourning that touches the entire country and far beyond," Parmelin said on X.
with DPA and AP
