  We investigate a number of coloring problems restricted to bipartite graphs
with bounded diameter. First, we investigate the $k$-List Coloring, List
$k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with
diameter at most $d$, proving NP-completeness in most cases, and leaving open
only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$.
  Some of these results are obtained through a proof that the Surjective
$C_6$-Homomorphism problem is NP-complete on bipartite graphs with diameter at
most four. Although the latter result has been already proved [Vikas, 2017], we
present ours as an alternative simpler one. As a byproduct, we also get that
$3$-Biclique Partition is NP-complete. An attempt to prove this result was
presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a
flaw in their proof, which we identify and discuss here.
  Finally, we prove that the $3$-Fall Coloring problem is NP-complete on
bipartite graphs with diameter at most four, and prove that NP-completeness for
diameter three would also imply NP-completeness of $3$-Precoloring Extension on
diameter three, thus closing the previously mentioned open cases. This would
also answer a question posed in [Kratochv\'il, Tuza, and Voigt, 2002].
