I haven't received any substantial responses to a similar question on math.stackexchange, so let me try here.
Let $R$ be a profinite ring (that is a projective limit of finite rings). Assume that the ring $R$ is local, of maximal ideal $m$, and assume that $m$ is open for the pro-finite topology. Then is $R$ necessarily $m$-adically separated ? $m$-adically complete?