  We are concerned with the three dimensional incompressible Navier--Stokes
equations driven by an additive stochastic forcing of trace class. First, for
every divergence free initial condition in $L^{2}$ we establish existence of
infinitely many global-in-time probabilistically strong and analytically weak
solutions, solving one of the open problems in the field. This result in
particular implies non-uniqueness in law. Second, we prove non-uniqueness of
the associated Markov processes in a suitably chosen class of analytically weak
solutions satisfying a relaxed form of an energy inequality. Translated to the
deterministic setting, we obtain non-uniqueness of the associated semiflows.
