  We report evidence for a new class of variable star, which we dub
millimagnitude RR Lyrae (mmRR). From K2 observations of the globular cluster
M4, we find that out of 24 horizontal branch stars not previously known to be
RR Lyrae variables, two show photometric variability with periods and shapes
consistent with those of first overtone RR Lyrae variables. The variability of
these two stars, however, have amplitudes of only one part in a thousand, which
is ${\sim}$200 times smaller than for any RR Lyrae variable in the cluster, and
much smaller than any known RR Lyrae variable generally. The periods and
amplitudes are: 0.33190704 d with 1.0 mmag amplitude and 0.31673414 d with 0.3
mmag amplitude. The stars lie just outside the instability strip, one blueward
and one redward. The star redward of the instability strip also exhibits
significant multi-periodic variability at lower frequencies. We examine
potential blend scenarios and argue that they are all either physically
implausible or highly improbable. Stars such as these are likely to shed
valuable light on many aspects of stellar physics, including the mechanism(s)
that set amplitudes of RR Lyrae variables.
