Completeness and reflexivity type properties of $B_1(X)$

For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given.