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.)
Let me give an example for your first question for $p=2$ and $r=3$, hoping I fully understand what you are looking for.
Consider in $\mathbb{P}^3$ the hypersurface defined by $z^2+zt+xu$ in characteristic $2$. By checking the partial derivatives after localizing in every chart, it is smooth.
Now to the plane: note that if you localize at $t=1$, you get the spectrum of $k[z,x,u]/(z^2+z+xu)$. If you now intersect with any $u=\alpha$ hyperplane, then you get the spectrum of $k[z,x]/(z^2+z+\alpha x)$ which is just the most basic Artin-Schreier cover of $\mathbb{A}^1$, and is known to be again (isomorph to) an affine line.
I am not sure how to generalize the construction to general characteristic and dimensions, even though I expect it to do.
