Although the bulk of the research in privacy and statistical disclosure control is designed for static data, more and more data are often collected as continuous streams, and extensions of popular privacy tools and models have been proposed for this scenario. However, most of these proposals require buffers, where incoming individuals are momentarily stored, anonymized, and then released following a delay, thus considering a data stream as a succession of batches while it is by nature continuous. Having a delay unavoidably alters data freshness but also, more critically, inordinately exerts constraints on what can be achieved in terms of protection and information preservation. By considering randomized response, and specifically its recent bistochastic extension, in the context of dynamic data, this paper proposes a protocol for the anonymization of data streams that achieves zero delay while exhibiting formal privacy guarantees. Using a new tool in the privacy literature that introduces the concept of elementary plausible deniability, we show that it is feasible to achieve an atomic processing of individuals entering a stream, in-stead of proceeding by batches. We illustrate the application of the proposed approach by an empirical example.