  Let $k$ be a perfect field of characteristic $p$ and $\Gamma$ an infinite,
first countable pro-$p$ group. We study the behavior of the $p$-primary part of
the "motivic class group", i.e. the full $p$-divisible group of the Jacobian,
in any $\Gamma$-tower of function fields over $k$ that is unramified outside a
finite (possibly empty) set of places $\Sigma$, and totally ramified at every
place of $\Sigma$. When $\Sigma=\emptyset$ and $\Gamma$ is a torsion free
$p$-adic Lie group, we obtain asymptotic formulae which show that the
$p$-torsion class group schemes grow in a remarkably regular manner. In the
ramified setting $\Sigma\neq\emptyset$, we obtain a similar asymptotic formula
for the $p$-torsion in "physical class groups", i.e. the $k$-rational points of
the Jacobian, which generalizes the work of Mazur and Wiles, who studied the
case $\Gamma=\mathbf{Z}_p$.
