Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, cream the softened unsalted butter and granulated sugar together until light and fluffy.
Add the eggs one at a time to the butter and sugar mixture, beating well after each addition.
Stir in the vanilla extract.
Mix in the mashed ripe bananas and sour cream until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until incorporated.
Pour the batter into the prepared baking pan and spread evenly.
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 allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack and cool completely before slicing and serving.