The four-storey building in Los Cristianos, on the south of the island, collapsed at 09:30 local time (08:30 GMT), emergency officials said.
The body of a woman aged about 70 was found in the rubble, local media said.
Residents living near the building said they had heard a large explosion shortly before it collapsed.
Three people were also injured in the incident, one seriously.
Nearby apartments were later evacuated over fears of a gas leak.
Arona municipality said searches for nine of the 28 people who lived in the building were continuing.
It is not clear if those who are unaccounted for were in the building when it fell. Mayor Jose Julian Mena told Spain's Efe news agency it was unclear whether anyone was trapped.
Emergency officials on the Canary Islands said (in Spanish) that a 57-year-old woman had suffered serious injuries to her upper body.
Two others, a 28-year-old Italian man and a 55-year-old woman, were also treated, the latter after a panic attack.
One nearby resident told Spain's Efe news agency there had been a strong smell of gas in the area soon after the explosion.
The owner of the Hide Away Pub, a bar next to the affected building, said his terrace had been "obliterated" by the collapse.
Javier Velazquez told British newspaper The Sun that if the blast had happened in the evening, the number of casualties would have been "far worse".
English-language newspapers on the island say the area is popular with British tourists.
One person has died and nine others remain missing after an apartment block collapsed on the Spanish island of Tenerife.