In a paper I am reading at the moment (Hrushovski Martin, elimination of imaginaries in $Q_p$), in some proof they use the following fact (at least this would be enough to get their proof going, but maybe we have more hypothesis) :
If we have a group $G$ acting on a set $X$ containing elements $(e_i\mid i<\omega)$ such that all $g\in G$ fixes all the $e_i$ but a finite number, is it true that all but a finite number of $e_i$ are fixed by $G$?
The authors say it is a consquence of Neumann's lemma (my guess is the one saying that a group covered by a finite number of cosets of subgroups is covered by the ones of finite index) but I have been trying to figure it out for some time already and cannot. If anyone has any idea of how that might work, a little help would be very welcome.
Some days later : Thanks a lot for the counter-examples, they really helped me understand.
If anyone is interested, what I really needed was that there are at most a finite number of $e_i$ with an infinite orbit under $G$. This is false in general (see the counter-example given in the answer) but Newmann's lemma does imply that if there are an infinity of elements with an infinite orbit then for all $N$, we can find a $g$ and $N$ elements that are not fixed by $g$. And as it happens that my $G$ is in fact the automorphism group of a very saturated model (sorry, some model theory had to barge in) by compacity I can find an automorphism that moves an infinity of $e_i$.
I hope any of this made sense.