An ordinary abelian variety with an étale self-isogeny of $p$-power degree and no isotrivial factors

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 etale self-isogeny p-power degree. As consequence, we deduce there exist varieties fields whose groups points the maximal purely inseparable extension is not finitely generated, answering in negative question Thomas Scanlon.