This study concentrates on the problem of robust quantised control for a class of switched non-strict-feedback non-linear systems under arbitrary switching. Differs from non-switched non-linear systems, the stability in switched ones (especially under arbitrary switching) is much dependent on the construction of common Lyapunov function (CLF). However, such CLF is fairly difficult to be designed in this study as the simultaneous consideration of full-states uncertainties, unknown virtual control coefficients, as well as quantisation problem. To overcome the challenge, this study presents a systematic control design method. Specifically, the neural networks with minimal learning parameter, and the bounded ness property of Gaussian basis functions are combined to deal with the full-states uncertainties, while an online estimator is built by sufficiently using the special structure of virtual controllers to dispose the unknown virtual coefficients problem. Moreover, a non-linear decomposition of quantiser is further proposed, rendering that the quantisation problem can be solved independently at the last step of backstepping iteration. With these results, an adaptive control algorithm which guarantees a CLF for all subsystems is successfully established such that the tracking error is steered into an adjustable area of origin asymptotically, and meanwhile all closed-loop signals remain uniformly ultimately bounded under arbitrary switching. Lastly, the obtained conclusions are well verified by the simulated results.