  We prove that a valued field of positive characteristic $p$ that has only
finitely many distinct Artin-Schreier extensions (which is a property of
infinite NTP$_2$ fields) is dense in its perfect hull. As a consequence, it is
a deeply ramified field and has $p$-divisible value group and perfect residue
field. Further, we prove a partial analogue for valued fields of mixed
characteristic and observe an open problem about 1-units in this setting.
Finally, we fill a gap that occurred in a proof in an earlier paper in which we
first introduced a classification of Artin-Schreier defect extensions.
