I have a reference or counterexample request. Suppose $k$ is a field and $\ell\neq char(k)$. There are several common references that show that $H^i_{et}(-, \mathbb{Q}_\ell )$ is a Weil cohomology theory when $k$ is separably closed.

I've spent several hours skimming through Milne's Etale Cohomology, the 1994 Motives volume, SGA articles, online searches, etc and I can't seem to determine whether or not $\ell$-adic cohomology forms a Weil cohomology theory when you don't assume you are in some "geometric" situation by making assumptions on the field.

Is there a reference that proves this is still a Weil cohomology theory or is it just false in this case? Thanks.

(This might be in SGA somewhere, but my skimming of French is rather slow and any specific related statement I find tends to throw in being over an algebraically closed field.)

Edit/Update: Everyone is commenting on the non-finitely generatedness, so I'll be more specific. That isn't really the interesting thing to me. Do you still have some sort of cycle class map that behaves nicely (functorially)? For instance, that paper Timo listed seems to imply that as long as finiteness is satisfied when you plug in a particular variety, everything else seems to be fine, but I haven't had time to seriously look at it yet.