Eight refugees with partial burns due to the blaze, which erupted in a house within a cluster, were sent to a government hospital in Noakhali district, Bhasan Char police chief Kawsar Alam Bhuiyan said, Reuters reported.
He said five children were among the injured.
Bangladesh has relocated around 32,000 people from border camps in the southeastern district of Cox's Bazar to Bhasan Char Island since late 2020.
The move has faced opposition, especially from aid groups worried about a disaster in a country that regularly faces severe weather, especially along its coast.
The government has dismissed safety concerns over the island, citing the building of flood defenses as well as housing for 100,000 people, hospitals and cyclone centres.
Nearly a million members of the Muslim minority from Myanmar live in crammed, bamboo-and-plastic camps in Cox's Bazar, most of them having fled a military crackdown in 2017.
Fires often break out in the crowded camps with their makeshift structures. A massive blaze in March 2021 killed at least 15 refugees and destroyed more than 10,000 homes.
Last year about 12,000 were left homeless after nearly 2,800 shelters and more than 90 facilities including hospitals and learning centres were destroyed in a fire. A panel that investigated the blaze called it a "planned act of sabotage".