  In this paper we introduce and study Bernstein spaces on a class of quadratic
CR manifolds, that we call Siegel CR manifolds. These are spaces of entire
functions of exponential type whose restrictions to a given Siegel CR
submanifold are $L^p$-integrable with respect to a natural measure. For these
spaces, among other results, we prove the Plancherel-P\'olya inequality, a
Bernstein inequality and a sufficient condition for a sequence to be sampling.
