Well I'm confused, say that you take $X=\mathbf{C}/\Lambda$ with $\Lambda=\mathbf{Z}+i\mathbf{Z}$ and you take the isometry $z\mapsto z+\frac{1}{2}$, then there is no fixed point! Are you assuming that your space is simply connected?

Well $K(GL(n,Z/p),1)=X$ will be a CW-complex of infinite dimension and the only way I know to construct it is with the usual killing cells technique which is kind of tautological

@Fernando, do you have a reference? So I looked quickly at Dieudonne's book and I found the definition of pseudomanifold but I could not find the result.

I should have thought about it! Thanks!

What is a good reference which explaines why $G_{\mathbf{Q}_p}$ is topologically finitely generated?

So how do you show the existence of such a basis of neighborhoods at $e$ in $H$. Definitely you need $H$ to be closed but I don't see how to use it.

well you just added in your statement the assumption that $G$ is locally euclidean, so now it is fine!

Thanks a lot Andreas, this is exactly what I was looking for.