  We apply Malliavin calculus to the $\Phi^4_3$ equation on the torus and prove
existence of densities for the solution of the equation evaluated at regular
enough test functions. We work in the framework of regularity structures and
rely on Besov-type spaces of modelled distributions in order to prove Malliavin
differentiability of the solution. Our result applies to a large family of
Gaussian space-time noises including white noise, in particular the noise may
be degenerate as long as it is sufficiently rough on small scales.
