I am looking for a characterization of topological spaces $X$ whose function spaces $C_k(X,\mathbb R)$ and $C_k(X,\{0,1\})$ are Baire.

Here for topological spaces $X,Y$ by $C_k(X,Y)$ we denote the space of continuous functions from $X$ to $Y$, endowed with the compact-open topology.

It is well-know that for any complete metric space $Y$ and and compact space $X$ the space $C_k(X,Y)$ is metrizable by the complete metric $d(f,g):=\max_{x\in X}d_Y(f(x),g(x))$. 

This implies that for a $k_\omega$-space $X$ and any Polish space $Y$ the function space $C_k(X,Y)$ is Cech-complete and hence Baire.

**Problem 1.** Is there a $k$-space $X$ whose function space $C_k(X,\mathbb R)$ or $C_k(X,\{0,1\})$ is Baire but not Cech-complete?

**Problem 2.** Let $X$ be a zero-dimensional $k$-space (containing a unique non-isolated point). Are the following two statements equivalent?

1. $C_k(X,\mathbb R)$ is Baire;

2. $C_k(X,\{0,1\})$ is Baire.

I suspect that such questions should be considered in the literature. Could you give me any references?