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.