Is there an infinite, countable connected $T_2$space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(x) = x$.)

1$\begingroup$ I'd have tried with the Golomb space minus some lacunary subset (maybe $\{n!:n\ge 1\}$ is OK). Just to discard those maps $n\mapsto kn$ and remain connected. $\endgroup$ – YCor Nov 19 '17 at 20:23

$\begingroup$ Can you elaborate a bit and maybe add this as an answer? Thanks @YCor $\endgroup$ – Dominic van der Zypen Nov 19 '17 at 20:28

1$\begingroup$ No, because I'm far from a proof. It would require a proof of the fixed point property. I only suspect it holds. $\endgroup$ – YCor Nov 19 '17 at 21:29

$\begingroup$ About continuous selfmaps of $\mathbb{G}$, see also mathoverflow.net/a/286575/14094 $\endgroup$ – YCor Nov 21 '17 at 12:59
Yes, there is such an example.
This is nearly Problem 10705 in The American Mathematical Monthly, proposed by D. W. Brown in 106 #1 (January 1999), p. 67, where the problem asks for a countably infinite $T_{2}$ example. In an editorial comment following John Cobb's solution in 107 #4 (April 2000), pp. 375376, it is stated that Prabir Roy's lattice space  A countable connected Urysohn space with a dispersion point, Duke Mathematical Journal 33 #2 (1966), pp. 331333  is an example that is a countable connected Urysohn space. For what it's worth, my notes on this problem say that this can be verified by making use of the analog in Roy's space of observations (a) and (b) at the beginning of the proof of the theorem at the bottom of p. 375.