  In this article we consider some questions raised by F. Benoist, E. Bouscaren
and A. Pillay. We prove that infinitely $p$-divisible points on abelian
varieties defined over function fields of transcendence degree one over a
finite field are necessarily torsion points. We also prove that when the
endomorphism ring of the abelian variety is $\mZ$ then there are no infinitely
$p$-divisible points of order a power of $p$.
