If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point.

However for general spaces it is well-known that just because a space is contractible, it does not follow that it strongly deformation retracts to a point. Here I am using the Wikipedia terminology. An example can be found in Hatcher's "Algebraic Topology" (pg18) (see this question). But of course this example is a bit pathological.

I am wondering just how bad these examples have to be?

A nice class of spaces which is much more general that CW-complexes, but discards many pathological examples is the class of $\Delta$-generated spaces. For example the $\Delta$-generation of the rationals or the cantor sets are just discrete sets. Such spaces disappear in the $\Delta$-generated category, and so does Hatcher's counter example.

Is there an example of a contractible $\Delta$-generated space which fails to deformation retract onto a point?