  We show that a magnetic field in the high-symmetry direction lifts the
macroscopic classical ground-state degeneracy of the honeycomb $\Gamma$ model
and induces a long-range magnetic order. While a simple spin-polarized state is
stabilized for the ferromagnetic $\Gamma$-exchange, a periodic $\sqrt{3}\times
\sqrt{3}$ magnetic order is selected by magnetic field for the
antiferromagnetic interaction. We show that the complex spin structure of the
tripled unit cell can be described by the magnetization vector and a N\'eel
order parameter, similar to those for the spin-flop state of a bipartite
antiferromagnet. Indeed, the transition from the low-field plaquette-ordered
spin liquid to the field-induced magnetic order can be viewed as a generalized
spin-flop transition. An accidental O(2) degeneracy associated with rotation
symmetry of the N\'eel vector is broken by either quantum or thermal
fluctuations, leaving a six-fold degenerate ground state. At high fields, the
breaking of the ground-state $Z_6$ symmetry is through two
Berezinskii-Kosterlitz-Thouless transitions that enclose a critical XY phase.
