  We prove local Lipschitz regularity for weak solutions to a class of
degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u=
\sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $\Omega\times
\R^+$, where $\Omega$ is domain in the Heisenberg group $\Hn$, and $2\le p \le
4$. The result continues to hold in the more general setting of contact
sub-Riemannian manifolds.
