Definable types in algebraically closed valued fields

In [15][D. Marker, 1994], Marker and Steinhorn characterized models M ≺ N of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models M ≺ N for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M are definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any n ≥ 1 that there is a pair M ≺ N of algebraically closed valued fields such that all n ‐types over M realized in N are definable but there is an n + 1 ‐type over M realized in N which is not definable.