We construct, for every prime p, a function field K of characteristic
p and an ordinary abelian variety A over K, with no isotrivial factors, that
admits an ´etale self-isogeny φ : A → A of p-power degree. As a consequence, we deduce that there exist ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.