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. 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.

There exists singular Fano varieties of dimension $n\ge 3$ in characteristic $p$ ($p$ small when compared to $n$) 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. 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.

There exists singular Fano varieties of dimension $n$ 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 Non-rational hypersurfaces. It is unclear to me if in any of his examples $X$ is actually smooth.

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. 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.