There is a nice generalization of Belyi's theorem in positive characteristic, proved by M. Saïdi in his paper *Revêtements modérés et groupe fondamental de graphe de groupes.* (Compositio Math. 107 (1997), no. 3), Théorème 5.6:

Let $C$ be a smooth projective curve defined over a field $K$ of characteristic $p>2$. The following conditions are equivalent:

 - The curve $C$ can be defined over $\bar{\mathbf F}_p$,
 - There exists a finite cover $C\to\mathbf P^1$ *tamely* ramified above $\infty,0$ and $1$ (and étale elsewhere).

The proof, very short and elegant, relies on a result of Fulton on the existence of covers with only double ramification. Unfortunately, the argument does not apply in characterisctic $2$, for which the question is still open (some recent progress in this situation can be found in S. Schröer's paper *Curves with only triple ramification*, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7).