Preheat oven to 350°F (175°C). Grease an 8-inch square baking pan.
Whisk together the flour, oats, baking powder, and salt. Set aside.
Using an electric mixer on medium speed, beat the butter, sugar, and brown sugar until light and fluffy. Add the egg and vanilla, and mix well. Mix in the bananas and peanut butter, mixing just until combined.
Reduce mixer speed to low. Gradually add the flour mixture, mixing just until combined.
Transfer the batter to the prepared pan and spread evenly. Bake 40-45 minutes, or until edges are browned and a pick inserted into the center comes out clean. Cool in the pan before cutting into bars.
Notes
Store in an airtight container at room temperature up to 3 or 4 days.