We consider the Cauchy problem for a spherically symmetric SU (2) Yang–Mills field propagating outside the Schwarzschild black hole. Although solutions starting from smooth finite energy initial data remain smooth for all times, not all of them scatter since there are non-generic solutions which asymptotically tend towards unstable static solutions. We show that a static solution with one unstable mode appears as an intermediate attractor in the evolution of initial data near a border between basins of attraction of two different vacuum states. We study the saddle-point dynamics near this attractor; in particular, we identify the universal phases of evolution: the ringdown approach, the exponential departure and the eventual decay to one of the vacuum states.