  We develop a theory of local densities and tangent cones in a motivic
framework, extending work by Cluckers-Comte-Loeser about $p$-adic local
density. We prove some results about geometry of definable sets in Henselian
valued fields of characteristic zero, both in semi-algebraic and subanalytic
languages, and study Lipschitz continuous maps between such sets. We prove
existence of regular stratifications satisfying analogous of Verdier condition
$(w_f)$. Using Cluckers-Loeser theory of motivic integration, we define a
notion of motivic local density with values in the Grothendieck ring of the
theory of the residue sorts. We then prove the existence of a distinguished
tangent cone and that one can compute the local density on this cone endowed
with appropriate motivic multiplicities. As an application we prove a
uniformity theorem for $p$-adic local density.
