Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake 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 3-4 minutes.
Add the eggs one at a time to the butter and sugar mixture, beating well after each addition.
Stir in the vanilla extract.
Alternately add the dry flour mixture and the buttermilk to the wet ingredients, beginning and ending with the flour mixture. Mix until just combined.
Fold in the mashed bananas gently until evenly incorporated.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Turn the cake out onto a wire rack and let it cool completely before serving.