In some clinical trials with two treatment arms, the patients enter the study at different times and are then allocated to one of two treatment groups. It is important for ethical reasons that there is greater probability of allocating a patient to the group that has displayed more favourable responses up to the patient's entry time. There are many adaptive designs in the literature to meet this ethical constraint, but most have a single binary response. Often the binary response is longitudinal in nature, being observed repeatedly over different monitoring times. This paper develops a randomized longitudinal play-the-winner design for such binary responses which meets the ethical constraint. Some performance characteristics of this design have been studied. It has been implemented in a trial of pulsed electro-magnetic field therapy with rheumatoid arthritis patients.