Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ?

More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of metrizable spaces, and the case of finite topological spaces ?

For example, is the following true:

Each endomorphism of a contractible quasi-compact $T_0$ space necessarily has a fixed point.

Note that being $T_0$ is necessary: the indiscrete space with two points is quasi-compact and contractible, yet the permutation of the two points has no fixed point. Perhaps one can relax the notion of a fixed point, e.g. by requiring that $x$ and $f(x)$ are topologically indistinguishable.

By the Brouwer theorem for finite topological spaces I mean what is implied by the Brouwer fixed point theorem for the geometric realisation of a finite topological space: namely, an endomorphism $f:K\to K$ of a finite topological space $K$ has a fixed point provided the endomorphism $|f|: |K|\to |K|$ of the geometric realisation of $K$ has a fixed point.