Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let 

$A$ be an ordinary abelian variety over $K$. Suppose that the N\'eron model $\CA$ of $A$ over $S$ has some closed fibre $\CA_s$, which is 

an abelian variety of $p$-rank $0$. We show that in this situation the group $A(K^\perf)$ is finitely generated (thus generalizing a special case of the Lang-N\'eron theorem). Here $K^\perf=K^{p^{-\infty}}$ is the maximal purely inseparable extension of $K$. This result  implies in particular that the "full" Mordell-Lang conjecture is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai. 

When $A$ is an elliptic curve, this result was proven by D. Ghioca using a different method