Does Fodor's lemma fail for countable ordinals?

For the usual statement of Fodor's lemma to make sense, one needs well-behaved notions of club and stationary sets, which fail for countable ordinals, so let me be more precise.

Question: Let $\alpha$ be a countable limit ordinal. Does there exist a weakly increasing, regressive function $f: \alpha \to \alpha$ such that the image of $f$ is cofinal in $\alpha$?

Here "regressive" means "$f(\beta) < \beta$ for all $0 < \beta < \alpha$". I'm also interested in the version of the question where $f$ is defined only on limit ordinals $<\alpha$.