  It is known that distillation in continuous variable resource theories is
impossible when restricted to Gaussian states and operations. To overcome this
limitation, we enlarge the theories to include convex mixtures of Gaussian
states and operations. This extension is operationally well-motivated since
classical randomness is easily accessible. We find that resource distillation
becomes possible for convex Gaussian resource theories-albeit in a limited
fashion. We derive this limitation by studying the convex roof extension of a
Gaussian resource measure and then go on to show that our bound is tight by
means of example protocols for the distillation of squeezing and entanglement.
