Free curves in Fano hypersurfaces must have high degree

The purpose of this note is to show that the minimal $e$ for which every smooth Fano hypersurface of dimension $n$ contains a free rational curve of degree at most $e$ cannot be bounded by a linear function in $n$ when the base field has positive characteristic. This is done by providing a super-linear bound on the minimal possible degree of a free curve in certain Fermat hypersurfaces.