Melt butter in a medium skillet over low heat. Add brown sugar and cinnamon; stir frequently and cook until sugar dissolves and sauce is thick, about 3 minutes. Add banana liqueur.

Slice bananas in half lengthwise and gently place in the skillet. Cook for 1 minute, then gently turn and cook until brown and soft, about 1 to 2 minutes more.

Add coffee-flavored liqueur and cook until hot, 1 to 2 minutes. Tip the skillet slightly and ignite the rum with a lighter. Remove skillet from the heat.

Scoop ice cream into 4 bowls. Add a banana to each bowl and pour warm sauce over top. Serve immediately.