We investigate several aspects of solutions to stochastic evolution equations in Hilbert spaces driven by a standard symmetric $α$-stable cylindrical noise. Similarly to cylindrical Brownian motion or Gaussian white noise, standard symmetric $α$-stable noise exists only in a generalised sense in Hilbert spaces. The main results of this work are the existence of a mild solution, long-term regularity of the solutions via Lyapunov functional approach, and an Itô formula for mild solutions to evolution equations under consideration. The main tools for establishing these results are Yosida approximations and an Itô formula for Hilbert space-valued semi-martingales where the martingale part is represented as an integral driven by cylindrical $α$-stable noise. While these tools are standard in stochastic analysis, due to the cylindrical nature of our noise, their application requires completely novel arguments and techniques.