Stir the coconut milk thoroughly so the thick cream and liquid are evenly combined before measuring.
In a large bowl, whisk together the flour, sugar, baking powder, and salt.
Add ¾ cup coconut milk and stir until a rough dough forms. If dry flour remains and the dough will not come together, add more coconut milk 1 tablespoon at a time. Stop when the dough is soft and slightly tacky but manageable; you may not need the full extra ¼ cup.
Transfer the dough to a lightly floured surface. Knead gently for 1–2 minutes, just until evenly combined and reasonably smooth. Avoid adding large amounts of flour or kneading until stiff.
Shape the dough into a ball, return it to the bowl, and cover. Let rest at room temperature for 30 minutes.
Divide the rested dough into 12 equal pieces. Roll each piece into a ball, then flatten and roll into a round approximately 3½ inches across and ¼ inch thick. If a round springs back, let it relax for a few minutes before rolling again.
Place the rounds on a lightly floured tray and keep covered while preparing the oil.
Pour oil into a heavy, deep-sided pan to a depth of approximately 1 inch, keeping the pan no more than half full. Heat to 350°F (175°C). Set a wire rack over a baking sheet nearby.
Carefully lower one dough round into the oil for a test batch. Fry for approximately 2–3 minutes per side, turning carefully, until golden and cooked through. The arepa may puff, but the amount of puffing will vary.
Transfer the test arepa to the rack. Let it cool briefly, then cut it open. The center should look cooked and bread-like, with no wet or gummy dough. If the outside browns before the center cooks, reduce the heat slightly and check that the remaining rounds are no thicker than ¼ inch.
Fry the remaining rounds in small batches without crowding the pan, maintaining the oil near 350°F (175°C). Allow the oil to return to temperature between batches.