  We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit
of Beardon's framework that unifies several results in this area. In
particular, it applies to strictly convex bounded domains in $\mathbb{R}^{n}$
or $\mathbb{C}^{n}$ with respect to a large class of metrics including
Hilbert's and Kobayashi's metrics. The results are generalized to $1$-Lipschitz
compact mappings in infinite-dimesional Banach spaces.
