I think p-adic fields themselves are somewhat chimeric. Although I know better, I can never fully avert the tendency to think of them as having characteristic p, rather than zero. Indeed, I just heard a number theorist refer to them as being of "mixed characteristic", meaning that although $\mathbb{Z}_p$ has characteristic zero, its residue field is $\mathbb{F}_p$ has characteristic p. I understand that this allows you to pass information from the Galois groups of finite fields (whose elements can be explicitly identified using Frobenius maps that only make sense in positive characteristic) to Galois groups of local fields, and thence to Galois groups of global fields.

Other bizarre characteristic-jumping arguments include Ax's proof of the Ax–Grothendieck theorem (an injective polynomial map is bijective), which reduces to varieties over finite fields by a logical compactness argument. There is the BBD (Beilinson–Bernstein–Deligne, secretly plus Gabber) proof of the Decomposition Theorem, which used weights of l-adic sheaves on schemes over finite fields to prove a theorem true only on complex varieties. And I have heard that Mori did...something...using an argument of this sort, but perhaps someone else could tell me what it was?

Basically, even though I know only a smattering of facts along these lines, I think you can find a whole zoo of finite-characteristic chimeras.