Spin liquids are exotic phases of quantum matter that challenge Landau’s paradigm of symmetry-breaking phase transitions. Despite strong exchange interactions, spins do not order or freeze down to zero temperature. Although well established for one-dimensional quantum antiferromagnets, in higher dimensions where quantum fluctuations are less acute, realizing and understanding such states is a major issue, both theoretically and experimentally. In this regard, the simplest nearest-neighbour Heisenberg antiferromagnet Hamiltonian on the highly frustrated kagome lattice has proven to be a fascinating and inspiring model. The exact nature of its ground state remains elusive and the existence of a spin-gap is the first key issue to be addressed to discriminate between the various classes of proposed spin liquids. Here, through low-temperature NMR contrast experiments on high-quality single crystals, we single out the kagome susceptibility and the corresponding dynamics in the kagome archetype, the mineral herbertsmithite, ZnCu3(OH)6Cl2. We firmly conclude that this material does not harbour any spin-gap, which restores a convergence with recent numerical results promoting a gapless Dirac spin liquid as the ground state of the Heisenberg kagome antiferromagnet. © 2020, The Author(s), under exclusive licence to Springer Nature Limited.