We characterize those Lie groups, and algebraic groups over a local field of
characteristic zero, whose first reduced L^p-cohomology is zero for all p>1,
extending a result of Pansu. As an application, we obtain a description of
Gromov-hyperbolic groups among those groups. In particular we prove that any
non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local
field of zero characteristic is quasi-isometric to a 3-regular tree. We also
extend the study to semidirect products of a general locally compact group by a
cyclic group acting by contracting automorphisms.