  A topological space $X$ is Baire if the intersection of any sequence of open
dense subsets of $X$ is dense in $X$. One of the interesting problems for the
space of Baire functions is the Banakh-Gabriyelyan problem: Let $\alpha$ be a
countable ordinal. Characterize topological spaces $X$ and $Y$ for which the
function space $B_{\alpha}(X,Y)$ is Baire. In this paper, for any Frechet space
$Y$ , we have obtained a characterization topological spaces $X$ for which the
function space $B_{\alpha}(X,Y)$ is Baire. In particular, we proved that
$B_{\alpha}(X,\mathbb{R})$ is Baire if and only if $B_{\alpha}(X,Y)$ is Baire
for any Banach space $Y$. Also we proved that many completeness and compactness
properties coincide in spaces $B_{\alpha}(X,Y)$ for any Frechet space $Y$.
