  In this paper together with the preceding Part I \cite{CHR}, we develop a
framework for tame geometry on Henselian valued fields of characteristic zero,
called Hensel minimality. It adds to \cite{CHR} the treatment of the mixed
characteristic case. Hensel minimality is inspired by o-minimality and its role
in real geometry and diophantine applications. We develop geometric results and
applications for Hensel minimal structures that were previously known only
under stronger or less axiomatic assumptions, and which often have counterparts
in o-minimal structures. We prove a Jacobian property, a strong form of Taylor
approximations of definable functions, resplendency results and cell
decomposition, all under Hensel minimality, more precisely, $1$-h-minimality.
We obtain a diophantine application of counting rational points of bounded
height on Hensel minimal curves.
