sorry, age does that.. No way to avoid it. You can delay it a bit by making sure your bananas aren't showing bare above the pudding and you refrigerate it sealed while not eating.. Still, it's a natural process.
When I make banana pudding, I make sure my gang comes by. They devour it so fast the bananas don't have time to turn brown.
Sorry, that's the best answer I have. As OP says, chill it but time will naturally cause the bananas to turn brown.
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Other people asked questions on various topics, and are still waiting for answer. Would be great if you can take a sec and answer them