  We construct solutions of higher-dimensional Einstein gravity coupled to
nonlinear $\sigma$-model with cosmological constant. The $\sigma$-model can be
perceived as exterior configuration of a spontaneously-broken $SO(D-1)$ global
higher-codimensional "monopole". Here we allow the kinetic term of the
$\sigma$-model to be noncanonical; in particular we specifically study a
quadratic-power-law type. This is some possible higher-dimensional
generalization of the Bariola-Vilenkin (BV) solutions with $k$-global monopole
studied recently. The solutions can be perceived as the exterior solution of a
black hole swallowing up noncanonical global defects. Even in the absence of
comological constant its surrounding spacetime is asymptotically non-flat; it
suffers from deficit solid angle. We discuss the corresponding horizons. For
$\Lambda>0$ in $4d$ there can exist three extremal conditions (the cold,
ultracold, and Nariai black holes), while in higher-than-four dimensions the
extremal black hole is only Nariai. For $\Lambda<0$ we only have black hole
solutions with one horizon, save for the $4d$ case where there can exist two
horizons. We give constraints on the mass and the symmetry-breaking scale for
the existence of all the extremal cases. In addition, we also obtain factorized
solutions, whose topology is the direct product of two-dimensional spaces of
constant curvature ($M_2$, $dS_2$, or $AdS_2$) with (D-2)-sphere. We study all
possible factorized channels.
