  We consider weak solutions to very singular parabolic equations involving a
one-Laplace-type operator, which is singular and degenerate, and a
$p$-Laplace-type operator with $\frac{2n}{n+2}<p<\infty$, where $n\ge 2$
denotes the space dimension. This type of equation is used to describe the
motion of a Bingham flow. It has been a long-standing open problem of whether a
spatial gradient of a weak solution is continuous in space and time. This paper
aims to give an affirmative answer for a wide class of such equations. This
equation becomes no longer uniformly parabolic near a facet, the place where a
spatial gradient vanishes. To achieve our goal, we show local a priori
H\"{o}lder continuity of gradients suitably truncated near facets. For this
purpose, we consider a parabolic approximate problem and appeal to standard
methods, including De Giorgi's truncation and comparisons with Dirichlet heat
flows. Our method is a parabolic adjustment of our method developed to prove
the corresponding statements for stationary problems.
