Combine the butter, sugar, and cinnamon in a flambé pan or skillet.
Place the pan over low heat either on an alcohol burner or on top of the stove, and cook, stirring, until the sugar dissolves.
Stir in half of the rum, and then place the bananas in the pan.
When the banana pieces soften and begin to brown, carefully add the remaining rum.
Continue to cook the sauce until the rum is hot, and then tip the pan slightly to ignite the rum.
When the flames subside, divide the bananas and the sauce evenly atop the ice cream portions.