We demonstrate the use of a multidimensional extension of the latent Markov model to analyse data from studies with repeated binary responses in developmental psychology. In particular, we consider an experiment based on a battery of tests which was administered to pre-school children, at three time periods, in order to measure their inhibitory control (IC) and attentional flexibility (AF) abilities. Our model represents these abilities by two latent traits which are associated to each state of a latent Markov chain. The conditional distribution of the test outcomes given the latent process depends on these abilities through a multidimensional one-parameter or two-parameter logistic parameterisation. We outline an EM algorithm for likelihood inference on the model parameters; we also focus on likelihood ratio testing of hypotheses on the dimensionality of the model and on the transition matrices of the latent process. Through the approach based on the proposed model, we find evidence that supports that IC and AF can be conceptualised as distinct constructs. Furthermore, we outline developmental aspects of participants' performance on these abilities based on inspection of the estimated transition matrices. (Contains 7 tables and 2 figures.)