We extend the renormalisation operator introduced in \cite{dCML} from
period-doubling H\'enon-like maps to H\'enon-like maps with arbitrary
stationary combinatorics. We show the renormalisation picture holds also holds
in this case if the maps are taken to be \emph{strongly dissipative}. We study
infinitely renormalisable maps $F$ and show they have an invariant Cantor set
$\mathcal{O}$ on which $F$ acts like a $p$-adic adding machine for some $p>1$.
We then show, as for the period-doubling case in \cite{dCML}, the sequence of
renormalisations have a universal form, but the invariant Cantor set
$\mathcal{O}$ is non-rigid. We also show $\mathcal{O}$ cannot possess a
continuous invariant line field.