  We are concerned with the three dimensional incompressible Navier--Stokes
equations driven by an additive stochastic forcing. 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.
