Emergency workers managed to rescue 15 people when the roof caved in, but three later died of their injuries, local police chief Jahedur Rahman told AFP.
Rahman said up to 10 people remained trapped at the site in Mongla, the country's second largest sea port.
Buildings frequently collapse in Bangladesh owing largely to widespread use of shoddy building materials and a failure to abide by proper engineering and construction rules.
In April 2013 a nine-storey garment factory complex collapsed just outside the capital Dhaka leaving at least 1,138 people dead.
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