Preheat the oven to 350°F (175°C). Grease and flour a 9×9-inch square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition.
Stir in the vanilla extract.
Mix in the mashed bananas until combined.
Add the sour cream and mix until smooth.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Fold in the semi-sweet chocolate chips evenly throughout the batter.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan on a wire rack for 15 minutes.
After 15 minutes, remove the cake from the pan and allow it to cool completely on the wire rack before slicing and serving.