The ground state of the quantum kagome antiferromagnet Zn-brochantite,ZnCu$_3$(OH)$_6$SO$_4$, which is one of only a few known spin-liquid (SL)realizations in two or three dimensions, has been described as a gapless SLwith a spinon Fermi surface. Employing nuclear magnetic resonance in a broadmagnetic-field range down to millikelvin temperatures, we show that in appliedmagnetic fields this enigmatic state is intrinsically unstable against a SLwith a full or a partial gap. A similar instability of the gaplessFermi-surface SL was previously encountered in an organic triangular-latticeantiferromagnet, suggesting a common destabilization mechanism that most likelyarises from spinon pairing. A salient property of this instability is that aninfinitesimal field suffices to induce it, as predicted theoretically for someother types of gapless SL's.
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