With the rack in the middle position, preheat the oven to 180 °C (350 °F). Butter a 23 x 13-cm (9 x 5-inch) loaf pan (a 1.5-litre (6-cup) capacity pan) and line with a strip of parchment paper, letting it hang over both sides.
In a bowl, combine flour, baking powder and baking soda. Set aside.
In another bowl, combine bananas, coconut milk and lime juice. Set aside.
In a third bowl, cream butter and sugar with an electric mixer. Add egg and vanilla and beat until smooth. At low speed, add dry ingredients alternately with banana mixture.
Pour into the loaf pan. Bake for about 1 hour or until a toothpick inserted in the centre of the cake comes out clean.
Let cool on a wire rack before removing from the pan and slicing.