A Pila--Wilkie theorem for Hensel minimal curves

Recently, a new axiomatic framework for tameness in henselian valued fields was developed by Cluckers, Halupczok, Rideau-Kikuchi and Vermeulen and termed Hensel minimality. In many aspects, it aims to mimic o-minimality for non-archimedean fields. In this article we develop the first Diophantine applications of Hensel minimality. We prove a Pila--Wilkie type of theorem for transcendental curves definable in Hensel minimal structures. In order to do so, we introduce a new notion of point counting in this context related to dimension counting over the residue field. Finally, we give three examples showcasing the necessity of this new dimension counting.