Preheat oven to 350 degrees F (175 degrees C). Grease several baking sheets, or line with parchment paper.
Beat the butter and sugar together in a bowl until the mixture is soft and creamy. Beat in the egg until well blended. Stir in the cornmeal about 1/4 cup at a time, stirring to blend well after each addition, then stir in the flour, baking soda, lime zest, coconut, salt, and rum until the mixture forms a soft dough. Drop by rounded teaspoons onto the prepared baking sheets about 2 inches apart.
Bake in the preheated oven until the edges just turn golden, 8 to 12 minutes, and allow cookies to cool on the baking sheets before removing. Cookies are very fragile when warm.
