Preheat the oven to 350° F. Line one muffin pan with paper liners or butter and set aside.
In a bowl, whisk together flour, baking powder and salt and set aside.
In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and 1/3 cup sugar until light and fluffy, 3 to 4 minutes, scraping down sides of the bowl as needed.
Beat in the rum, then the banana. With the mixer on low speed, add the flour mixture in three parts, alternating with the milk, until just combined. Transfer mixture to a large bowl and set aside.
Rinse the bowl of the electric mixer and add the whisk attachment. Beat the egg whites on low speed until foamy. With the mixer running, gradually add the remaining 1/8 cup sugar and beat on high speed until stiff, glossy peaks form, about 4 minutes. Don’t overbeat these lovely peaks or they will turn lumpy.
Gently fold a third of the egg-white mixture into the butter-flour mixture until combined. Then gently fold in the rest.
Divide the batter evenly among the muffin cups. Bake, rotating pans halfway through, until the cupcakes are golden brown and a toothpick inserted in the center of a cupcake comes out clean, 25 to 30 minutes. Remove and let cool.