Preheat the oven to 350°F (175°C). Grease and flour an 8×8 inch baking pan.
In a large mixing bowl, cream together the softened butter and granulated sugar until light and fluffy.
Add the eggs one at a time, beating well after each addition, then stir in the vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking soda, and salt.
Add the dry ingredients to the creamed mixture alternately with the sour cream, beginning and ending with the flour mixture. Mix until just combined.
Fold in the mashed bananas gently until evenly distributed.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
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.
Once cooled, cut the cake into 8 equal slices and serve.