Combine butter, brown sugar, and cinnamon in a large skillet over medium heat; cook, stirring constantly, until butter melts and sugar dissolves. Add banana; cook 2 minutes, coating slices in sugar mixture. Remove from heat.
Heat rum in a small saucepan over medium-low heat. Pour heated rum into one side of skillet. Ignite rum with a long match; let flames die down. Return to medium heat; cook 30 seconds, stirring gently to coat banana.
Place 1/2 cup ice cream in each of 5 goblets, and top with 1/2 cup banana mixture. Serve immediately.