Preheat oven to 350°F (175°C). Lightly grease an 8-inch square baking pan.
Whisk together the flour, baking powder, and salt. Set aside.
Using an electric mixer on medium speed or mixing by hand, combine the brown sugar, sugar, butter, and banana. Add the eggs, one at a time, mixing well after each addition. Mix in the vanilla.
If using an electric mixer, reduce the mixer speed to low. Gradually mix in the flour mixture.
Stir in the chocolate chips.
Transfer the batter to the prepared pan, and spread evenly. Bake 30 to 35 minutes, or until the bars are browned and appear set, and a pick inserted into the center comes out clean.
Cool completely in the pan before cutting into bars.