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Normal in one base and not in another?

Examples of numbers that are normal in some bases are known (Champernowne's $0.12345678910111213\ldots$ is probably the most well-known example of a number that is normal in base 10).

To my knowledge (but please correct me if I'm wrong) no numbers that are normal (in all bases) are known.

But are any examples known of numbers that are normal in one base and not in some other base?