  We present novel analytic hairy black holes with a flat base manifold in the
(3+1)-dimensional Einstein SU(2)-Skyrme system with negative cosmological
constant. We also construct (3+1)-dimensional black strings in the Einstein
$SU(2)$-non linear sigma model theory with negative cosmological constant. The
geometry of these black strings is a three-dimensional charged BTZ black hole
times a line, without any warp factor. The thermodynamics of these
configurations (and its dependence on the discrete hairy parameter) is analyzed
in details. A very rich phase diagram emerges.
