  A general fixed point theorem for isometries in terms of metric functionals
is proved under the assumption of the existence of a conical bicombing. It is
new even for isometries of Banach spaces as well as for non-locally compact
CAT(0)-spaces and injective spaces. Examples of actions on non-proper
CAT(0)-spaces come from the study of diffeomorphism groups, birational
transformations, and compact K\"ahler manifolds. A special case of the fixed
point theorem provides a novel mean ergodic theorem that in the Hilbert space
case implies von Neumann's theorem. The theorem accommodates classically
fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach
and Prus. Moreover, from the main theorem together with some geometric
arguments of independent interest, one can deduce that every bounded invertible
operator of a Hilbert space admits a nontrivial invariant metric functional on
the space of positive operators. This is a result in the direction of the
invariant subspace problem although its full meaning is dependent on a future
determination of such metric functionals.
