I believe the "complete feather", a non-hausdorff $1$-manifold due to Haefliger and Reeb, is such an example. It is a simple generalization of Gabriel C. Drummond-Cole example of the interval with two endpoints. Actually, we discuss it in 

>Mathieu Baillif, Alexandre Gabard, _Manifolds: Hausdorffness versus homogeneity_, Proceedings of the American Mathematical Society **136** no 3 (2008) pp 1105–1111, doi:[10.1090/S0002-9939-07-09100-9](https://doi.org/10.1090/S0002-9939-07-09100-9), arXiv:[math/0609098](https://arxiv.org/abs/math/0609098).

(sorry for the self-promotion). This space is $\Delta$-generated, if I am not mistaken.

Say that a space $X$ is locally strongly contractible if each point $x\in X$ has a neighborhood which strongly deformation retracts to $x$. (The retraction needs to be defined only in the neighborhood, not the whole space.) David Gauld proved long ago the following theorem:
If a space $X$ is locally strongly contractible, contractible to a point $p$, and completely regular at $p$, then $X$ strongly deformation retracts to $p$. The complete feather shows that you cannot drop entirely the "completely regular" assumption.