Natural examples of deformation retracts that are not strong deformation retracts

This depends on what you mean by being "specifically constructed for this purpose". A nice example (that I'm sure inspired the example given by Hatcher) is the comb space $$X=[0,1]\times\{0\}\cup\{0\}\times [0,1]\cup\bigcup_{n=1}^\infty \left(\left\{\frac{1}{n}\right\}\times [0,1]\right)$$

Any connected subset of the leftmost tooth $\{0\}\times[0,1]$ is a deformation retract of $X$, since $X$ contracts to the point $(0,0)$. But most of the time these won't be strong deformation retracts; in fact the only subset of that tooth that is a strong deformation retract is the point $(0,0)$.

For completeness, I think the answer to this question is answered in Spanier's Algebraic Topology in Theorem 1.4.11 on p.31:

If $(X \times I, (X \times 0) \cup (A \times I) \cup (X \times 1))$ has the homotopy extension property with respect to $X$ and $A$ is closed in $X$, then $A$ is a deformation retract of $X$ if and only if $A$ is a strong deformation retract of $X$.