Preheat the oven to 350°F (175°C). Grease and flour an 8-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
In a large bowl, beat the softened butter and granulated sugar together until light and fluffy, about 3 minutes.
Add the eggs one at a time, beating well after each addition, then mix in the vanilla extract.
Stir in the mashed ripe bananas and sour cream until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until incorporated. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 35 to 40 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 for 10 minutes.
Turn the cake out onto a wire rack and allow it to cool completely before serving.