Heat oven to 375° F. Coat an 8½-by-4½-inch loaf pan with butter and dust with the flour; tap out the excess. Whisk together the flour, baking powder, baking soda, and salt in a large bowl; set aside.
In a separate bowl, whisk together the butter and sugar until no lumps remain.
Whisk in the eggs, vanilla, banana, and sour cream. Stir in the carrot and coconut. Stir the wet ingredients into the dry ingredients until just combined.
Transfer the batter to the prepared pan and bake until a toothpick inserted in the center of the loaf comes out clean, 65 to 70 minutes. Cool the bread in the pan for 30 minutes. Turn out the loaf on a wire rack to cool further. Serve warm or at room temperature. Or cool completely, wrap, and freeze for up to 2 weeks.
* Total time includes cooling