Four of the missing were crew on board a boat that sunk when the storm accompanied by strong winds and heavy rain hit the island in the Arabian Sea, Fisheries Resources Minister Fahad Kafin told AFP.
Four people on the boat were rescued while three other people were missing when floods swept away their vehicles, the minister said.
Around 150 families had been evacuated and moved to government facilities after downpours caused houses and streets to flood, trapping people in their homes, he said.
Rescue workers could not reach the southern and eastern parts of the island which they are located in the centre of the cyclone, Kafin said.
Some residents carrying children tried to escape through the flooded streets, an AFP correspondent said.
Authorities called on humanitarian organisations and the Saudi-led military coalition that is battling Huthi rebels in the country to help, according to state-run news agency Saba.
The internationally recognised government of President Abedrabbo Mansour Hadi declared early Thursday Socotra a "disaster province" due to severe damage, according to Saba.
"Socotra is a disaster province due to human and material damage at all levels and requires urgent aid," government spokesman Rajeh Badi said.
Hadi called Socotra governor Ramzy Mahrous and assured him of providing all available facilities and aid to help rescue affected people.
jpegMpeg4-1280x720Yemeni relief officials called on international aid agencies to "send medical teams urgently" to Socotra and other areas in South Yemen expected to be hit by the fast moving cyclone.
Millions of Yemenis are living in dire conditions as a result of a long-running civil conflict, which since 2015 has pitted a Saudi-led coalition against the Iran-backed Huthi rebels.
But Socotra has been spared involvement in the violence, which has claimed nearly 10,000 lives since March 2015 and triggered what the United Nations has called the world's worst humanitarian crisis.
In neighbouring Oman, authorities announced through the official news agency they were taking "necessary precautions" in case the cyclone hits the Gulf sultanate.
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