  The main objective of the present paper is to construct a new class of
space-time discretizations for the stochastic $p$-Stokes system and analyze its
stability and convergence properties.
  We derive regularity results for the approximation that are similar to the
natural regularity of solutions. One of the key arguments relies on discrete
extrapolation that allows to relate lower moments of discrete maximal
processes.
  We show that, if the generic spatial discretization is constraint conforming,
then the velocity approximation satisfies a best-approximation property in the
natural distance.
  Moreover, we present an example such that the resulting velocity
approximation converges with rate $1/2$ in time and $1$ in space towards the
(unknown) target velocity with respect to the natural distance. The theory is
corroborated by numerical experiments.
