  We study the existence of hairy black holes in the generalized
Einstein-Skyrme model. It is proven that in the BPS model limit there are no
hairy black hole solutions, although the model admits gravitating (and flat
space) solitons. Furthermore, we find strong evidence that a necessary
condition for the existence of black holes with Skyrmionic hair is the
inclusion of the Skyrme term $\mathcal{L}_4$. As an example, we show that there
are no hairy black holes in the $\mathcal{L}_2+\mathcal{L}_6+\mathcal{L}_0$
model and present a new kind of black hole solutions with compact Skyrmion hair
in the $\mathcal{L}_4+\mathcal{L}_6+\mathcal{L}_0$ model.
