Yukio Matsumoto in "A 4-manifold which admits no spine", see here, constructed a compact PL $4$-manifold with boundary that is homotopy equivalent to the $2$-torus but does not deformation retract to a PL-embedded copy of $T^2$.
There are also examples of this phenomenon in higher even dimensions by Cappell and Shaneson, see here.
I do not know whether these manifolds admit topological (i.e. non PL) spines.