  A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a
semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a
non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a
connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove
that $G$ is normal in the full automorphism group ${\rm Aut}(\Gamma)$ of
$\Gamma$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(\Gamma)$. As an
application, we classify half-arc-transitive bipartite bi-Cayley graphs over
the group $G$ of valency less than $2p$. Furthermore, it is shown that there
are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the
group $G$ of valency less than $p$.
