On algebraically maximal valued fields that are not defectless

An example originally given by F.~Delon shows the existence of an algebraically maximal discretely valued field of characteristic $p>0$ which admits purely inseparable extensions of degree $p^2$ with defect $p$. These extensions are not generated by a single element. Using a trick introduced in an earlier paper of the author, we construct algebraically maximal valued fields, of characteristic $p$ as well as of characteristic $0$, which admit separable extensions of degree $p^2$ with defect $p$. They are of rank 2 and it is an open question whether such examples having rank 1 exist.