This article investigates the problem of global stabilisation for a class of switched nonlinear systems in p-normal form whose subsystems are not assumed to be asymptotically stabilisable. Unlike the existing results on systems in p-normal form where the power order is only positive odd integer, we allow positive even integer of the power order. Using the convex combination method and the adding a power integrator technique, we construct a switching law and design state-feedback controllers of individual subsystems explicitly by a recursive design algorithm to guarantee asymptotic stability of the closed-loop system. The designed method is also extended to the global stabilisation problem of switched nonlinear systems in p-normal form with zero-dynamics. As an application of the proposed design method, a continuously stirred tank reactor is studied.