  We study the parabolic $p$-Laplacian system in a bounded domain. We deduce
optimal convergence rates for the space-time discretization based on an
implicit Euler scheme in time. Our estimates are expressed in terms of
Nikolskii spaces and therefore cover situations when the (gradient of) the
solution has only fractional derivatives in space and time. The main novelty is
that, different to all previous results, we do not assume any coupling
condition between the space and time resolution $h$ and $\tau$. The theoretical
error analysis is complemented by numerical experiments.
