  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 SL
with a spinon Fermi surface. Employing nuclear magnetic resonance in a broad
magnetic-field range down to millikelvin temperatures, we show that in applied
magnetic fields this enigmatic state is intrinsically unstable against a SL
with a full or a partial gap. A similar instability of the gapless
Fermi-surface SL was previously encountered in an organic triangular-lattice
antiferromagnet, suggesting a common destabilization mechanism that most likely
arises from spinon pairing. A salient property of this instability is that an
infinitesimal field suffices to induce it, as predicted theoretically for some
other types of gapless SL's.
