Butter a deep ovenproof dish. Cut the crusts off the bread or leave them on if you prefer, then butter the slices and cut them in half on the diagonal. Arrange half the slices over the base of the dish, buttered-side up.
Sprinkle over half the banana slices and half the chocolate chips. Sprinkle over half the lime zest. Repeat these layers – bread, banana, chocolate chips and zest.
Whisk the sugar and eggs together until smooth, then stir in the cream, milk and rum. Pour this mixture over the contents of the dish, pressing down the bread lightly. Leave to stand for half an hour.
Preheat the oven to 180°C/Fan 160°C/Gas 4. Place the dish on a baking tray. Sprinkle the top of the pudding with demerara sugar, making sure all the bananas are covered with it, then bake in the oven for 35–40 minutes until the custard has set and slightly puffed up and the top is lightly browned.
Remove from the oven and leave to stand for 10 minutes before serving. Great hot or cold.