Preheat the oven to 350°F (175°C) and grease two 9-inch cake pans.
In a mixing bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.
In a separate mixing bowl, beat the butter and granulated sugar together until creamy.
Beat in the eggs, one at a time, until fully incorporated.
Mix in the vanilla extract and mashed bananas until fully combined.
Add the dry ingredients to the wet ingredients, mixing until just combined.
Mix in the sour cream until just combined.
Divide the cake batter evenly between the two prepared cake pans.
Bake the cakes for 25-30 minutes, or until a toothpick inserted into the center of the cakes comes out clean.
Allow the cakes to cool completely in the pans.
In a large saucepan, combine the brown sugar, unsalted butter, and heavy cream.
Cook the mixture over medium heat until the sugar dissolves, stirring constantly.
Increase the heat to medium-high and cook for an additional 2-3 minutes, or until the mixture thickens.
Remove the caramel sauce from the heat and stir in the dark rum.
Place one cake layer on a serving plate and spread half of the caramel sauce on top.
Place the second cake layer on top of the caramel sauce.
Drizzle the remaining caramel sauce over the top of the cake.
Serve the bananas foster cake warm or at room temperature.