  We study the symmetric stochastic $p$-Stokes system, $p \in (1,\infty)$, in a
bounded domain. The results are two-folded.
  First, we show that in the context of analytically weak solutions the
stochastic pressure -- related to non-divergence free stochastic forces --
enjoys almost $-1/2$ temporal derivatives on a Besov scale.
  Second, we verify that the velocity component~$u$ of strong solutions obeys
$1/2$ temporal derivatives in an exponential Nikolskii space. Moreover, we
prove that the non-linear symmetric gradient $V(\varepsilon u) = (\kappa +
|\varepsilon u|)^{(p-2)/2} \varepsilon u$, $\kappa \geq 0$, has $1/2$ temporal
derivatives in a Nikolskii space.
