The properties of quantum materials are commonly tuned using experimental variables such as pressure, magnetic field and doping. Here we explore a different approach: irreversible, plastic deformation of single crystals. We show for the superconductor SrTiO$_3$ that compressive plastic deformation induces low-dimensional superconductivity significantly above the superconducting transition temperature ($T_c$) of undeformed samples, with evidence of superconducting correlations at temperatures two orders of magnitude above the bulk $T_c$. The superconductivity enhancement is correlated with the appearance of self-organized dislocation structures, as revealed by diffuse neutron and X-ray scattering. We also observe signatures of deformation-induced quantum-critical ferroelectric fluctuations and inhomogeneous ferroelectric order via Raman scattering. These results suggest that the strain surrounding the self-organized dislocation structures induces local ferroelectricity and quantum-critical dynamics that strongly influence $T_c$, consistent with a theory of superconductivity enhanced by soft polar fluctuations. More broadly, our results demonstrate the promise of plastic deformation and dislocation engineering as tools to manipulate electronic properties of quantum materials.