  We consider a spherically symmetric (magnetic) $SU(2)$ Yang-Mills field
propagating on the exterior of the extremal Reissner-Nordstr\"om black hole.
Taking advantage of the conformal symmetry, we reduce the problem to the study
of the Yang-Mills equation in a geodesically complete spacetime with two
asymptotically flat ends. We prove the existence of infinitely many static
solutions (two of which are found in closed form) and determine the spectrum of
their linear perturbations and quasinormal modes. Finally, using the
hyperboloidal approach to the initial value problem, we describe the process of
relaxation to the static endstates of evolution for various initial data.
