  Given a compact K\"ahler manifold, we prove that all global isometries of the
space of K\"ahler metrics are induced by biholomorphisms and
anti-biholomorphisms of the manifold. In particular, there exist no global
symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion
does not even admit local symmetries. Closely related to these findings, we
provide a large class of metric geodesic segments that can not be extended at
one end, pointing out the first such examples in the literature.
