We consider the regularity of solutions to a multidimensional moving boundary problem modelling the growth of non-necrotic solid tumours. The model equations include two elliptic equations describing the concentration of a nutrient and the distribution of the internal pressure within the tumour, respectively, and a first-order partial differential equation governing the evolution of the moving boundary on which surface tension effects counteract the internal pressure. On account of the moving boundary and surface tension effects, this problem is a nonlinear problem involving non-local terms. By employing the functional analytic method and the theory of maximal regularity, we prove that the moving boundary is real analytic in temporal and spatial variables, even if the given initial data admit less regularity.