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, about 3-4 minutes.
Add the eggs one at a time, beating well after each addition.
Mix in the mashed bananas, sour cream, and vanilla extract 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 in the preheated oven for 40-45 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 cool completely before serving.