  We study non-degenerate, totally umbilical surfaces of a special class of
pseudo-Riemannian manifolds, namely Walker three-manifolds. We show that such
surfaces are either one of a totally geodesic family described by Calvaruso and
Van der Veken or the ambient manifold must be locally conformally flat (here
the surface can also be totally geodesic). The proof makes use of a key
technique deployed by Manzano and Soaum in their recent classification of
totally umbilical surfaces in homogeneous Riemannian three-manifolds.
