  In this paper, we study the random walk on a supercritical branching process
with an uncountable and unbounded set of types supported on the $d$-regular
tree $\mathbb{T}_d$ ($d\geq 3$), namely the cluster $\mathcal{C}_\circ^h$ of
the root in the level set of the Gaussian Free Field (GFF) above an arbitrary
value $h\in (-\infty, h_{\star})$. The value $h_{\star}\in (0,\infty)$ is the
percolation threshold; in particular, $\mathcal{C}_\circ^h$ is infinite with
positive probability. We show that on $\mathcal{C}_\circ^h$ conditioned to be
infinite, the simple random walk is ballistic, and we give a law of large
numbers and a Donsker theorem for its speed. To do so, we design a renewal
construction that withstands the long-range dependencies in the structure of
the tree. This allows us to translate underlying ergodic properties of
$\mathcal{C}_\circ^h$ into regularity estimates for the random walk.
