The low-energy (epsilon = (h) over bar < 1 meV), low-temperature (T = 0.05 K) spin dynamics of the s = 1/2 kagome candidate herbertsmithite are probed in the presence of magnetic fields up to 2.5 T. The zero-field spectra reveal a very weak continuum of scattering at T = 10 K and a broad inelastic peak centred at epsilon(max) = 0.2 meV at lower temperatures, T < 1 K. The broad peak is found to be strongly damped, with a liquid-like structure factor implying correlations at length scales up to r = 6 angstrom. The field dependence of the peak appears to follow the Zeeman splitting of s = 1/2 excitations, consistent with the weakly split 'doublets' observed in low-temperature specific heat. A possible explanation of these observations is a short-range correlated state involving defect spins between the kagome planes and moments in the kagome layers.