Let $X$ be a Hausdorff topological space with the following property:
For every continuous function $f:X\to X$, there is a finite subset $S\neq \emptyset$ of $X$ with $F(S)\subset S$
Does this implies that: either $X$ is a finite set or $X$ has the fixed point property?
What about if $X$ is a manifold? Does the above condition implies FPP for $X$?
Edit: According to the interesting example of Alex to the previous version of question, we ask what about if we require $X$ to be connected?