  We consider $d\geqslant 4$ Einstein--(extended-)Yang-Mills theory, where the
gauge sector is augmented by higher order terms. Linearizing the (extended)
Yang-Mills equations on the background of the electric Reissner-Nordstr\"om
(RN) black hole, we show the existence of normalizable zero modes, dubbed
non-Abelian magnetic stationary clouds. The non-linear realization of these
clouds bifurcates the RN family into a branch of static, spherically symmetric,
electrically charged and asymptotically flat black holes with non-Abelian hair.
Generically, the hairy black holes are thermodynamically preferred over the RN
solution, which, in this model, becomes unstable against the formation of
non-Abelian hair, for sufficiently large values of the electric charge.
