This paper presents a new control method for a class of nonlinear switched systems in strict feedback form with a state constraint. In the proposed approach, a nonlinear mapping is first introduced to transform the problem of controlling the switched system with state constraint to a new problem of regulating the transformed system without a constraint. By using the backstepping technique, the designed state-feedback controller ensures that the corresponding closed-loop system is asymptotically stable without violation of the constraint, and that all closed-loop signals keep bounded. Furthermore, we explore the use of a coordinate transformation to deal with an asymmetric state constraint as a generalised approach. In the end, an application example on a chemical process is shown to demonstrate the efficacy of the obtained results.