We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz-Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy-Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones. An application to zero-sum stochastic games is also given.