  Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular
extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a
$K$-abelian variety such that all the isogeny factors are neither isotrivial
nor of $p$-rank zero. We give a necessary and sufficient condition for the
finite generation of $A(K^{perf})$ in terms of the action of $End(A)\otimes
\mathbb Q_p$ on the $p$-divisible group $A[p^{\infty}]$ of $A$. In particular
we prove that if $End(A)\otimes \mathbb Q_p$ is a division algebra then
$A(K^{perf})$ is finitely generated. This implies the "full" Mordell-Lang
conjecture for these abelian varieties. In addition we prove that all the
infinitely $p$-divisible elements in $A(K^{perf})$ are torsion. These reprove
and extend previous results to the non ordinary case. One of the main technical
intermediate result is an overconvergence theorem for the Dieudonn\'e module of
certain semiabelian schemes over smooth varieties.
