We present a first-principles study of the properties of ordinary hexagonal ice (phase Ih) and of its proton-ordered version (phase XI) under the action of static electric fields. We compute the mechanical response to the field in addition to the ionic current-voltage diagrams; we also analyze several other microscopic aspects of the proton transfer mechanism, with particular emphasis on the role played by the oxygen sublattice in driving molecular dissociation. We further study the topological aspects of the mechanical and electrical responses by orienting the external field along two different crystalline directions in both ice samples. At variance with ice Ih, ice XI displays an anisotropic behavior in the range of explored field intensities. In fact, when the direction of the field coincides with the ferroelectric axis, sustained molecular dissociation and proton transfer events are both observed just beyond a given field intensity; instead, the two processes exhibit different activation thresholds when the field is oriented along another symmetry axis. The underlying mechanism of molecular dissociation appears to be the same in solid and liquid water independently of the direction of the field.