  Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is,
two opposite arcs). In this paper, we consider the case where such graphs are
bipartite and in which the undirected and directed degrees are one. The best
graphs, in terms of the number of vertices, are presented for small diameters.
Moreover, two infinite families of such graphs with diameter $k$ and number of
vertices of the order of $2^{k/2}$ are proposed, one of them being totally
regular $(1,1)$-mixed graphs. In addition, we present two more infinite
families called chordal ring and chordal double ring mixed graphs, which are
bipartite and related to tessellations of the plane. Finally, we give an upper
bound that improves the Moore bound for bipartite mixed graphs for $r = z = 1$.
