One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count points (and if coefficients have $\mathrm{char}\:p$ you could only count $\mathrm{mod}\:p$, barring some inventiveness).
The question is: if we take the Wikipedia definition of Weil cohomology theory and keep all of the requirements but allow characteristic to be positive, will we get any new theories?
I am not very experienced with etale cohomology to see whether it taken with $F_p$-coefficients satisfies all of the requirements. Assuming it does, taking it is a kind of a cheat because $l$-adic cohomology has a natural integral structure and $F_p$-coefficients are obtained by quotienting out the maximal ideal. The same objection applies to crystalline cohomology with truncated coefficients.
I wonder whether there are examples "genuinely" of $\mathrm{char}\:p$, i.e. such that there is no Weil cohomology theory with $\mathrm{char}\:0$ coefficients that has a natural integral structure over a DVR such that the quotient by maximal ideal produces the cohomology theory under consideration. Possibly there are some other more-or-less trivial "cheats" I am missing, point them out in the comments. An answer to this question should not rely on such "cheats" (the distinction is not very precise admittedly but I hope the idea is clear).