The Néron-Ogg-Shafarevich theorem usually seems to be cited to say that an abelian variety over a finite extension $K/{Q}_p$ has good reduction at $\ell \neq p$ if and only if the associated $\ell$-adic Galois representation (Tate module) is unramified, as seen in Serre-Tate. I am aware that these conditions can be slightly generalized (cf. the Néron models book). I also know that you get the analogous result at $\ell = p$ if you replace "unramified" by "crystalline," but that this was proven much more recently.
However, from skimming the proofs of these various results, I don't immediately see where the fact that $K$ is characteristic zero is used. For instance, Serre-Tate doesn't seem to say anything about the characteristic of $K$. Nevertheless, I haven't yet seen this theorem applied when $K$ has positive characteristic, and some of the results (e.g. the Coleman-Iovita paper where it is shown if the p-adic Tate module is crystalline, then the abelian variety has good reduction) do assume that $K$ is characteristic zero (and perfect residue field, but I am less concerned about this).
I guess I have a couple of related questions:
Does the Néron-Ogg-Shafarevich theorem for abelian varieties hold when $K$ has positive characteristic? If so, could you link to a paper where it is applied in this fashion?
Can you get similar results "at $p$" when char$(K) = p > 0$?
Yes, it is true in the function field setting. This will follow from the Neron models argument in Serre--Tate.
Note that "crystalline" only makes sense in the mixed char. setting. If you work directly with $p$-divisible groups (i.e. talk about the $p$-divisible group extending over the ring of integers) then the result should be true in the char. $p$ setting. My feeling is that this kind of thing would be proved in the part of SGA7 dealing with Neron models. (E.g. an extension of the Neron mapping to $p$-divisible groups is proved in there somewhere, although I forget the precise statement or hypotheses.) I'm always made a bit nervous about non-perfect residue fields too, so let me not commit to anything on that point, since I'm just writing off the top of my head.