  We consider a branching Markov process in continuous time in which the
particles evolve independently as spectrally negative L\'evy processes. When
the branching mechanism is critical or subcritical, the process will eventually
die and we may define its overall maximum, i.e. the maximum location ever
reached by a particule. The purpose of this paper is to give asymptotic
estimates for the survival function of this maximum. In particular, we show
that in the critical case the asymptotics is polynomial when the underlying
L\'evy process oscillates or drifts towards $+\infty$, and is exponential when
it drifts towards $-\infty$.
