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 soda, and salt. Set aside.
In a large bowl, cream the softened butter and granulated sugar together until light and fluffy using a hand mixer or stand 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 ingredients and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Pour the batter into the prepared cake 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.
Turn the cake out onto a wire rack and let it cool completely before serving.