We explore the small mass limit of a stochastic wave equation (SWE) driven by cylindrical $α$-stable noise, where $α\in (1,2)$, and prove that it converges to a stochastic heat equation. We establish its well-posedness, and in particular, the càdlàg property, which is not trivial in the infinite dimensional case. Using a splitting technique, we decompose the velocity component into three parts, which gives convenience to the moment estimate. We show the tightness of solution of SWE by verifying the infinite dimensional version of Aldous condition. After these preparation, we pass the limit and derive the approximation equation.