  We establish global-in-time existence and non-uniqueness of probabilistically
strong solutions to the three dimensional Navier--Stokes system driven by
space-time white noise. In this setting, solutions are expected to have space
regularity at most $-1/2-\kappa$ for any $\kappa>0$. Consequently, the
convective term is ill-defined analytically and probabilistic renormalization
is required. Up to now, only local well-posedness has been known. With the help
of paracontrolled calculus we decompose the system in a way which makes it
amenable to convex integration. By a careful analysis of the regularity of each
term, we develop an iterative procedure which yields global non-unique
probabilistically strong paracontrolled solutions.Our result applies to any
divergence free initial condition in $L^{2}\cup B^{-1+\kappa}_{\infty,\infty}$,
$\kappa>0$, and implies also non-uniqueness in law.
