Examples of geometric Bombieri-Lang-Noguchi outside Mordell-Lang: Nonrigid varieties with ample but not globally generated cotangent bundle

In this note we provide examples of nonisotrivial varieties over a function fields with ample cotangent bundle and zero Albanese. This provides explicit examples of varieties which are not embeddable into Abelian varieties for which the geometric Bombieri-Lang-Noguchi conjecture is known to be true.