  We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By
symmetric we mean that the full gradient of $p$-Laplacian is replaced by its
symmetric part, which causes breakdown of the Uhlenbeck structure. We derive
the interior regularity of time derivatives of its local weak solution. To
circumvent the space-time growth mismatch, we devise a new local regularity
technique of iterations in Nikolskii-Bochner spaces. It is interesting by
itself, as it may be modified to provide new regularity results for the
full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having
the regularity result for time derivatives, we obtain respective regularity of
the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems
on their embeddings and interpolations, may be of independent interest.
