Infinite "almost rigid" homogeneous $T_2$-space

Topological groups are homogeneous. In

J. van Mill, "A topological group having no homeomorphisms other than translations," Transactions of the AMS 280 (1983), pp. 491-498 (link),

Jan van Mill constructed an infinite topological group whose only self-homeomorphisms are group translations. Such a space is called "uniquely homogeneous" -- it is homogeneous, but for any pair of points there is exactly one self-homeomorphisms of the space witnessing homogeneity. Jan's group also has the amazing property that removing any point results in a rigid space.

In the same paper (section 4), van Mill shows that, assuming the Continuum Hypothesis, there is a topological group whose only continuous self-maps are either group translations or constant functions.

Thus the answer to your question is "consistently yes, and you can come close in ZFC." I do not know whether anyone else has come along and improved Jan's CH result to a ZFC result (but a quick glance through the papers citing Jan's seems to indicate that no one has).

Remark. This answer was written before the "infinite" assumption was added.

Take the discrete space with two points.

It is clearly homogeneous and Hausdorff, and its only self-maps are the two constant maps, the identity and the involution exchanging the two points.