Beat butter at medium speed with an electric mixer until creamy. Gradually add powdered sugar, beating until mixture is smooth. Stir in coconut and vanilla.
Stir together flour, baking powder, and salt. Gradually add to butter mixture, beating at low speed until blended.
Shape dough into 3 (10-inch) logs; wrap logs separately in wax paper, and chill 4 hours. Cut each log into 28 slices, and place slices 1 inch apart on lightly greased baking sheets.
Bake at 350° for 10 to 12 minutes or until edges are golden. Cool on wire racks.