Preheat the oven to 350°F. Generously grease a 13x9-inch baking pan with butter then set it aside.
In a large bowl, whisk together the milk, coconut milk, eggs, sugar, vanilla extract and mashed bananas.
Add the cubed bread and chocolate chips to the bowl and fold until the bread is well coated with the mixture.
Scoop the bread mixture into the prepared baking pan, spreading it into an even layer.
Bake the bread pudding, uncovered, until fully cooked and a toothpick inserted comes out clean, 30 to 35 minutes. Remove the bread pudding from the oven, let it cool for at least 10 minutes then slice and serve.