The singularly perturbed reaction-diffusion problem is considered on the unit square with homogenous Dirichlet boundary conditions. Its solution typically contains boundary layers and corner layers. It is discretised by a finite element method that uses rectangular Morley elements on a Shishkin mesh. In an associated energy-type norm that is natural for this problem, we prove an $O(\eps^{1/2}N^{-1}+\eps N^{-1}\ln N + N^{-3/2})$ rate of convergence for the error in the computed solution, where $N$~is the number of mesh intervals in each coordinate direction. Thus in the most troublesome regime when $\eps \approx N^{-1}$, our method is proved to attain an $O(N^{-3/2})$ rate of convergence, which is shown to be sharp by our numerical experiments and is superior to the $O(N^{-1/2})$ rate that is proved in Meng \& Stynes, Adv. Comput. Math. 2019 when Adini finite elements are used to solve the same problem on the same mesh.