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 our small elementary paper with A. Gabard (sorry for the self-promotion). This space is $\Delta$-generated, if I am not mistaken.

Say that a point is locally strongly contractible if each point has a neighborhood which strongly deformation retracts to a point. (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.

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, arXiv: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.