  The study of the ultrarelativistic plasmas in perturbation theory is plagued
with infrared divergences which are not eliminated by the screening
corrections. They affect, in particular, the computation of the lifetime of the
elementary excitations, thus casting doubt on the validity of the quasiparticle
picture. We show that, for Abelian plasmas at least, the infrared problem of
the damping rate can be solved by a non-perturbative treatment based on the
Bloch-Nordsieck approximation. The resulting expression of the fermion
propagator is free of divergences, and exhibits a {\it non-exponential} damping
at large times: $S_R(t)\sim \exp\{-\alpha T t \ln\omega_pt\}$, where
$\omega_p=gT/3$ is the plasma frequency and $\alpha=g^2/4\pi$.
