Peel bananas; cut each banana in half lengthwise. Cut each half into 2 pieces.
Melt butter in a large nonstick skillet over medium heat. Stir in brown sugar, liqueur, and cinnamon. Bring to a simmer, and cook for 2 minutes. Add bananas; cook for 4 minutes or until tender. Remove from heat. Add rum to pan, and ignite rum with a long match. Stir bananas gently until flame dies down. Serve over ice cream.