For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second notation, although it is not very correct),

In characteristic 0, as far as I know, there is a classification. $X$ has to be isomporphic either to $\mathbb{P}^1\times \mathbb{P}^1$ or a blow-up of $\mathbb{P}^2$ at $9-K_X^2\geq 0$ points in general position, i.e. not 3 points in the same line and not 6 points in the same conic of $\mathbb{P}^2$.

I would like to know if there is such a classification in characteristic $p\geq 2$. As far as I know the classification is definitely the same for $p>3$, and probably even for $p=3$ and there are 'extra' surfaces for $p=2$.

The perfect answer would confirm whether these assertions are true, describe the extra cases (in particular in terms of which curves can live in them or minimality) and/or give a reference.

But anything is better than nothing, so even if you know a bit of this I would like to know.