Over an algebraically closed field of characteristic $p$, does there exist a smooth degree $p$ hypersurface $X\subset\mathbb{P}^r$ that contains a plane $\Gamma$ of dimension $\lfloor \frac{r-1}{2}\rfloor$?

What about a smooth degree $p$ hypersurface $X$ such that a hyperplane section $X\cap H$ is a cone (considered as a subscheme of $H\cong \mathbb{P}^{r-1}$)?

(If we don't allow the degree to be divisible by $p$, then the Fermat hypersurface is an example of both.)