Probably, the discrete $\{0,1\}$ is not the counterexample Dominic van der Zypen expected to see :)

A more elaborate CH-example of a AFPP but not strongly rigid space was constructed by van Mill:

**Theorem 4.1 (van Mill, 1983)**. Under Continuum Hypothesis there exists a non-trivial metrizable separable connected locally connected Boolean topological group $X$ such that each continuous self-map $f:G\to G$ is either a translation or a constant map.

The space $G$ has AFFP but is not strongly rigid.

**Remark.** In the van Mill's proof the Continuum Hypothesis is used in combination with the following classical result of Sierpiński:

**Theorem (Sierpiński, 1921).** For any countable partition of the unit interval into closed subsets exactly one set of the partition is non-empty.

Motivated by this Sierpiński Theorem we can ask about the smallest infinite cardinality $\acute{\mathfrak n}$ of a partition of the unit interval into closed non-empty subsets. It is clear that $\acute{\mathfrak n}\le\mathfrak c$. The Sierpinski Theorem guarantees that $\omega_1\le\acute{\mathfrak n}$. This inequality can be improved to $\mathfrak d\le\acute{\mathfrak n}\le\mathfrak c$.

It seems that the proof of van Mill's Theorem actually yields more:

**Theorem**. Under $\acute{\mathfrak n}=\mathfrak c$ (which follows from $\mathfrak d=\mathfrak c$) there exists a non-trivial metrizable separable connected locally connected Boolean topological group $X$ such that each continuous self-map $f:G\to G$ is either a translation or a constant map.