In clinical studies, patients are usually accrued sequentially. Response-adaptive designs are then useful tools for assigning treatments to incoming patients as a function of the treatment responses observed thus far. In this regard, doubly adaptive biased coin designs have advantageous properties under the assumption that their responses can be obtained immediately after testing. However, it is a common occurrence that responses are observed only after a certain period of time. The authors examine the effect of delayed responses on doubly adaptive biased coin designs and derive some of their asymptotic properties. It turns out that these designs are relatively insensitive to delayed responses under widely satisfied conditions. This is illustrated with a simulation study.