  An exact hairy asymptotically locally AdS black hole solution with a flat
horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is
constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere
and depends explicitly on Killing coordinates, but in such a way that its
energy-momentum tensor is compatible with a metric with Killing fields. The
solution is characterized by a discrete parameter which has neither topological
nor Noether charge associated with it and therefore represents a hair. A $U(1)$
gauge field interacting with Einstein gravity can also be included. The
thermodynamics is analyzed. Interestingly, the hairy black hole is always
thermodynamically favored with respect to the corresponding black hole with
vanishing Pionic field.
