There exists singular Fano varieties of dimension $n$ in characteristic $p$ with a non-zero section of $\Omega^{n-1}_X\otimes \mathcal L^*$ for an ample $\mathcal L$.
The existence of these examples is established in Kollár’s paper [Nonrational hypersurfaces][1]. They are constructed as degree $p$ coverings of Fano manifolds ramified over smooth hypersurfaces. It is unclear to me if in any of his examples $X$ is actually smooth.


  [1]: https://www.ams.org/journals/jams/1995-08-01/S0894-0347-1995-1273416-8/S0894-0347-1995-1273416-8.pdf