We have used the HIRES echelle spectrograph on the Keck I telescope to obtain high-resolution spectroscopy of 51 late-type M dwarfs in the Hyades cluster. Cross-correlating the calibrated data against spectra of white dwarfs allows us to determine heliocentric velocities with an accuracy of +/- 0.3 km s(-1). 27 stars were observed at two epochs in 1997; two stars, RHy 42 and RHy 403, are confirmed spectroscopic binaries. RHy 42 is a double-lined, equal-mass system; RHy 403 is a single-lined, short-period binary, P similar to 1.275 d. RHy 403A has an absolute magnitude of M-I=10.85, consistent with a mass of 0.15 M.. The systemic mass function has a value [M-2 sin(i)](3)/(M-1+M-2)(2)=0.0085, which, combined with the non-detection of a secondary peak in the cross-correlation function, implies 0.095 > M-2> 0.07 M., and the strong possibility that the companion is the first Hyades brown dwarf to be identified. Unfortunately, the maximum expected angular separation in the system is only similar to 0.25 mas. Five other low-mass Hyads are identified as possible spectroscopic binaries, based either on repeat observations or on a comparison between the observed radial velocity and the value expected for Hyades cluster members. Combined with HST imaging data, we infer a binary fraction between 23 and 30 per cent. All of the stars are chromospherically active. RHy 281 was caught in mid-flare and, based on that detection, we estimate a flaring frequency of similar to 2.5 per cent for low-mass Hyades stars. Nine stars have rotational velocities, v sin(i), exceeding 20 km s(-1), and most of the sample have detectable rotation. We examine the H alpha emission characteristics of low-mass cluster members, and show that there is no evidence for a correlation with rotation.