The provincial governor’s office said another six people were injured when the snow hit a village in Shekay district, near the Tajikistan border.
The village had been swept away, said spokesman Abdul Marouf Rasikh, and the death toll is expected to rise.
Badakhshan is one of the country’s poorest and most remote regions and is shut off by heavy snow every year.
Afghanistan is suffering one of its harshest winters in many years.
Nasir Hemat, director of the Red Crescent in Badakhsham, said rescue teams had reached the remote site.
“There were 190 people living in the village – 39 people have been killed, 6 injured,” he told the BBC.
“Our rescue work is going on. We are hoping we can rescue a lot of people and prevent a human catastrophe.”
Mr Hemat said three people had also died in a nearby district.
Mr Rasikh told the BBC that provincial governor Shah Wali Ullah had been visiting at the time the avalanche hit on Monday night.
He was rescued by helicopter and taken to a remote area on the border with Tajikistan, he said.
Mr Hemat said 60 people had been killed by the snow in Badakhshan this year. and homes and cattle had been lost.
Up to 4m (13ft) of snow was lying in some areas of the province, he said, and roads between the capital, Faizabad, and remote rural areas were impassable for six months of the year.
“It has been a tragedy this year,” he said.