= Solution
Take \b[the <natural numbers> as the <dense> <discrete topology>[discrete subspace] of their <Stone-Čech compactification> $\beta\mathbb N$]. The ambient <compact Hausdorff space> is normal, and the <Stone-Čech extension property> makes this subspace $C^b$-embedded. It is not $C$-embedded: the <continuous function> $n\mapsto n$ is unbounded, whereas every <continuous> <real-valued function> on a <compact space> is bounded. The subspace is not closed, so this does not contradict the preceding closed-subspace theorem.
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