  We prove that a Fano variety (with arbitrary singularities) of dimension $n$
in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri
constant of the anti-canonical divisor at some smooth point is greater than $n$
and classify Fano varieties whose anti-canonical divisors have Seshadri
constants $n$. In characteristic $p>5$ and dimension $3$, we also show that
Fano varieties $X$ with Seshadri constants $\epsilon(-K_X,x)>2+\epsilon$ at
some smooth point $x\in X$ (for some fixed $\epsilon>0$) have bounded
anti-canonical degrees.
