A space $(X,\tau)$ is called *rigid* if $\textrm{Aut}(X)=\{\textrm{id}_X\}$. We say $(X,\tau)$ is *strongly rigid* if for every continuous map $f:X\to X$ we have that $f = \textrm{id}_X$ or $f$ is constant (that is there is $x_0\in X$ such that $f(x)=x_0$ for all $x\in X$).

Is there a strongly rigid Hausdorff space with more than 1 element?