  We provide a construction of the moduli space of stable coherent sheaves in
the world of non-archimedean geometry, where we use the notion of Berkovich
non-archimedean analytic spaces. The motivation for our construction is Tony
Yue Yu's non-archimedean enumerative geometry in Gromov-Witten theory. The
construction of the moduli space of stable sheaves using Berkovich analytic
spaces will give rise to the non-archimedean version of Donaldson-Thomas
invariants.
  In this paper we give the moduli construction over a non-archimedean field
$\kk$. We use the machinery of formal schemes, that is, we define and construct
the formal moduli stack of (semi)-stable coherent sheaves over a discrete
valuation ring $R$, and taking generic fiber we get the non-archimedean
analytic moduli of semistable coherent sheaves over the fractional
non-archimedean field $\kk$. For a moduli space of stable sheaves of an
algebraic variety $X$ over an algebraically closed field $\kappa$, the
analytification of such a moduli space gives an example of the non-archimedean
moduli space.
  We generalize Joyce's $d$-critical scheme structure in \cite{Joyce} or
Kiem-Li's virtual critical manifolds in \cite{KL} to the world of formal
schemes, and Berkovich non-archimedean analytic spaces. As an application, we
provide a proof for the motivic localization formula for a $d$-critical
non-archimedean $\kk$-analytic space using global motive of vanishing cycles
and motivic integration on oriented formal $d$-critical schemes. This
generalizes Maulik's motivic localization formula for motivic Donaldson-Thomas
invariants.
