Preheat the oven to 350°F (175°C). Grease and flour a 9x9 inch baking dish.

In a mixing bowl, whisk together flour, baking powder, and sugar.

In another bowl, cream the butter and add eggs, vanilla, and milk. Mix well.

Combine the wet and dry ingredients, and pour into the prepared baking dish.

Bake for 20 minutes or until a toothpick comes out clean. Let cool.

In a bowl, combine sweetened condensed milk, evaporated milk, and whole milk.

Once the cake has cooled, poke holes all over the top and pour the tres leches mixture over it. Let it soak for at least 30 minutes.

In a bowl, whisk together the instant vanilla pudding mix and 2 cups of milk until thickened.

Layer sliced bananas over the soaked cake, then spread the pudding on top.

Top with whipped cream and additional banana slices if desired.