We show that if X is a complete metric space with uniform relative normal structure and G is a subgroup of the isometry group of X with bounded orbits, then there is a point in X fixed by every isometry in G. As a corollary, we obtain a theorem of U. Lang (2013) concerning injective metric spaces. A few applications of this theorem are given to the problems of inner derivations. In particular, we show that if L1(μ) is an essential Banach L1(G) -bimodule, then any continuous derivation δ:L1(G)→L∞(μ) is inner. This extends a theorem of B. E. Johnson (1991) asserting that the convolution algebra L1(G) is weakly amenable if G is a locally compact group.