  A promising route to realize entangled magnetic states combines geometrical
frustration with quantum-tunneling effects. Spin-ice materials are canonical
examples of frustration, and Ising spins in a transverse magnetic field are the
simplest many-body model of quantum tunneling. Here, we show that the tripod
kagome lattice material Ho${_3}$Mg${_2}$Sb${_3}$O${_{14}}$ unites an ice-like
magnetic degeneracy with quantum-tunneling terms generated by an intrinsic
splitting of the Ho$^{3+}$ ground-state doublet, which is further coupled to a
nuclear spin bath. Using neutron scattering and thermodynamic experiments, we
observe a symmetry-breaking transition at $T^{\ast}\approx0.32$ K to a
remarkable state with three peculiarities: a concurrent recovery of magnetic
entropy associated with the strongly coupled electronic and nuclear degrees of
freedom; a fragmentation of the spin into periodic and ice-like components; and
persistent inelastic magnetic excitations down to $T\approx0.12$ K. These
observations deviate from expectations of classical spin fragmentation on a
kagome lattice, but can be understood within a model of dipolar kagome ice
under a homogeneous transverse magnetic field, which we survey with exact
diagonalization on small clusters and mean-field calculations. In
Ho${_3}$Mg${_2}$Sb${_3}$O${_{14}}$, hyperfine interactions dramatically alter
the single-ion and collective properties, and suppress possible quantum
correlations, rendering the fragmentation with predominantly single-ion quantum
fluctuations. Our results highlight the crucial role played by hyperfine
interactions in frustrated quantum magnets, and motivate further investigations
of the role of quantum fluctuations on partially-ordered magnetic states.
