Preheat the oven to 350F/180 C and line a square 8x8 baking dish with parchment paper.
Stir the buttermilk and ground espresso together and set it aside.
Whisk together the flour, baking powder, and salt in a bowl and set it aside.
In a stand mixing bowl or with a hand mixer, cream the sugar and butter until light and fluffy, about 5 minutes.
Mix in the egg until combined.
Mix in the mashed banana and vanilla.
Alternate the dry ingredients and buttermilk by adding a third of the dry ingredients first, then half the buttermilk, and so on, mixing until each it almost combined.
Evenly spread the batter into the baking pan.
Bake for 30-40 minutes or until a toothpick inserted into the center comes out clean.
Let the cake cool on a wire rack.