Preheat your oven to 325°F and lightly butter, flour, and line a bread tin.
Beat together butter and brown sugar in a large bowl with an electric mixer at high speed until pale and fluffy. Beat in egg until combined, then add bananas, sour cream, and vanilla and beat until combined well. Reduce speed to low. Sieve in the flour, baking powder, baking soda and mix.
Spread the batter in the bread tin and bake until pale golden and a wooden pick or skewer inserted in center comes out clean, 20 to 25 minutes.
Cool in pan on a rack 10 minutes. Run a thin knife around edge of pan, then invert rack over cake and invert cake onto rack to cool completely.
While the cake is cooling, make the coconut frosting. Beat together cream cheese and butter in a bowl using cleaned beaters at medium speed until smooth. Reduce speed to low, then add confectioners sugar, cream of coconut and mix until combined. Increase speed to high and beat until light and fluffy, about 2 minutes.
Frost top of cooled cake and sprinkle with coconut flakes.