Combine milk and coconut in a bowl. Let it stand at room temperature for 30 minutes. In a medium sized bowl, combine all-purpose flour, baking powder, and baking soda. Stir to get ingredients well mixed. Peel and mash bananas in another bowl. Grease and flour an 8-inch x 8-inch square cake pan.
Preheat oven to 350˚F (180°C).
Cream butter and sugar until light and fluffy, about 2 minutes. Beat in egg. Add vanilla extract.
Then pour in half of the coconut mixture, half of the mashed bananas, and half of the flour mixture. Beat on low for 30 seconds. Add the remaining coconut mixture, mashed bananas, and flour mixture. Continue to beat until batter is smooth. This should take 1 to 2 minutes.
Pour batter into prepared pan.
Bake for 1 hour or until wooden pick inserted in the center comes out clean. Cool in the pan for 10 minutes, remove and cool on wire rack.
Cream Cheese Frosting
Beat together cream cheese, butter, and vanilla extract with wire whisk until light and fluffy, about 2 minutes. Gradually beat in powdered sugar until smooth and creamy, another 2 more minutes.
When cake is completely cool, spread frosting all over the cake.