Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan.
In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a large bowl, cream the softened butter and granulated sugar together until light and fluffy using an electric mixer.
Add the eggs one at a time, beating well after each addition.
Mix in the vanilla extract.
Add the mashed bananas and sour cream to the wet mixture and stir until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and allow the cake to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.