  We analyze the validity of a quasiparticle description of a superconducting
state at a metallic quantum-critical point (QCP). A normal state at a QCP is a
non-Fermi liquid with no coherent quasiparticles. A superconducting order gaps
out low-energy excitations, except for a sliver of states for non-s-wave gap
symmetry, and at a first glance, should restore a coherent quasiparticle
behavior. We argue that this does not necessarily hold as in some cases the
fermionic self-energy remains singular slightly above the gap edge. This
singularity gives rise to markedly non-BCS behavior of the density of states
and to broadening and eventual vanishing of the quasiparticle peak in the
spectral function. We analyze the set of quantum-critical models with an
effective dynamical 4-fermion interaction, mediated by a gapless boson at a
QCP, $V(\Omega) \propto 1/\Omega^\gamma$. We show that coherent quasiparticle
behavior in a superconducting state holds for $\gamma <1/2$, but breaks down
for larger $\gamma$. We discuss signatures of quasiparticle breakdown and
compare our results with the data.
