The blaze broke out on the 18th floor, fire official Selina Chowdhury said. Fifteen fire trucks responded, but the flames still spread quickly. Troops and police joined firefighters in the rescue work, said Abul Kalam Azad, spokesman for Prime Minister Sheikh Hasina.
An air force helicopter rescued three people, including the mall's chief security officer, from the roof, fire official Mahbubur Rahman said.
Rahman said the blaze was fanned by summer winds. It was "almost doused" nearly six hours after it started, police Chief Noor Mohammad said.
It was unclear how many people were inside the mall when the fire started.
Doctors at Shamarita Hospital said a security guard died from burns and another 17 people were being treated there for smoke inhalation. Other hospitals reported receiving at least 33 people suffering from smoke inhalation.
The cause of the fire was not immediately known. The floor where it began has the offices of the mall's owner, the Bashundhara company, which would normally be closed Friday, part of the weekend in Bangladesh. The mall opened in 2004.
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