  Fermi systems in the crossover regime between weakly coupled
Bardeen-Cooper-Schrieffer (BCS) and strongly coupled Bose-Einstein-condensate
(BEC) limits are among the most fascinating objects to study the behavior of an
assembly of strongly interacting particles. The physics of this crossover has
been of considerable interest both in the fields of condensed matter and
ultracold atoms. One of the most challenging issue in this regime is the effect
of large spin imbalance on a Fermi system under magnetic fields. Although
several exotic physical properties have been predicted theoretically, the
experimental realization of such an unusual superconducting state has not been
achieved so far. Here we show that pure single crystals of superconducting FeSe
offer the possibility to enter the previously unexplored realm where the three
energies, Fermi energy $\varepsilon_{\rm F}$, superconducting gap $\Delta$ and
Zeeman energy, become comparable. Through the superfluid response, transport,
thermoelectric response, and spectroscopic-imaging scanning tunneling
microscopy, we demonstrate that $\varepsilon_{\rm F}$ of FeSe is extremely
small, with the ratio $\Delta/\varepsilon_{\rm F}\sim1 (\sim0.3)$ in the
electron (hole) band. Moreover, thermal-conductivity measurements give evidence
of a distinct phase line below the upper critical field, where the Zeeman
energy becomes comparable to $\varepsilon_{\rm F}$ and $\Delta$. The
observation of this field-induced phase provides insights into previously
poorly understood aspects of the highly spin-polarized Fermi liquid in the
BCS-BEC crossover regime.
