Four of the missing were crew on board a boat that sunk when the storm hit the island in the Arabian Sea, a fisheries ministry official told AFP.
Another official said that 150 families had been evacuated and moved to government facilities after downpours caused houses and streets to flood, trapping people in their homes.
Saudi air defences destroy ballistic missile fired from Yemen
Some residents carrying children tried to escape through the flooded streets, an AFP correspondent said.
Yemen's Houthis and Saudi Arabia in secret talks to end war: sources
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.
In neighbouring Oman, authorities announced through the official news agency they were taking "necessary precautions" in case the cyclone hits the Gulf sultanate.
Pakistan assails Yemen's missile attacks targeting Saudi capital Riyadh
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.
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