Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first countable.
This makes me (naively) suspect that it is not Fréchet-Urysohn either, but I couldn't find anything that talks about this. Is it a Fréchet-Urysohn space?