Hölder gradient estimates for fractional $p$-caloric functions

We establish interior $C^{1,α}$ regularity estimates, for some $α> 0$, for fractional $p$-caloric functions when $p$ is in the range $p \in (2,2/(1-s))$. As a consequence, we deduce improved regularity in time, which yields interior continuous differentiability in time for $p \in [1/(1-s), 2/(1-s))$