To date, only 18 exoplanets with radial velocity (RV) semi-amplitudes <2m/s have had their masses directly constrained. The biggest obstacle to RV detection of such exoplanets is variability intrinsic to stars themselves, e.g. nuisance signals arising from surface magnetic activity such as rotating spots and plages, which can drown out or even mimic planetary RV signals. We use Kepler-37 - known to host three transiting planets, one of which, Kepler-37d, should be on the cusp of RV detectability with modern spectrographs - as a case study in disentangling planetary and stellar activity signals. We show how two different statistical techniques - one seeking to identify activity signals in stellar spectra, and another to model activity signals in extracted RVs and activity indicators - can enable detection of the hitherto elusive Kepler-37d. Moreover, we show that these two approaches can be complementary, and in combination, facilitate a definitive detection and precise characterisation of Kepler-37d. Its RV semi-amplitude of 1.22+/-0.31m/s (mass 5.4+/-1.4M_{Earth}) is formally consistent with TOI-178b's 1.05^+0.25^-0.30_m/s, the latter being the smallest detected RV signal of any transiting planet to date, though dynamical simulations suggest Kepler-37d's mass may be on the lower end of our 1{sigma} credible interval. Its consequent density is consistent with either a water-world or that of a gaseous envelope (~0.4% by mass) surrounding a rocky core. Based on RV modelling and a re-analysis of Kepler-37 TTVs, we also argue that the putative (non-transiting) planet Kepler-37e should probably be stripped of its 'confirmed' status.