  We prove under certain weak assumptions a black hole no-hair theorem in
spherically symmetric spacetimes for self-gravitating time-dependent multiple
scalar fields with an arbitrary target space admitting a Killing field with a
non-empty axis and arbitrary non-negative potential invariant under the flow of
the Killing field. It is shown that for such configurations the only
spherically symmetric and asymptotically flat black hole solutions consist of
the Schwarzschild metric and a constant multi-scalar map. In due course of the
proof we also unveil the intrinsic connection of the time-dependence of the
scalar fields with the symmetries of the target space.
