  We present resistivity and thermal-conductivity measurements of
superconducting FeSe in intense magnetic fields up to 35 T applied parallel to
the $ab$ plane. At low temperatures, the upper critical field $\mu_0
H_{c2}^{ab}$ shows an anomalous upturn, while thermal conductivity exhibits a
discontinuous jump at $\mu_0 H^{\ast}\approx 24$ T well below $\mu_0
H_{c2}^{ab}$, indicating a first-order phase transition in the superconducting
state. This demonstrates the emergence of a distinct field-induced
superconducting phase. Moreover, the broad resistive transition at high
temperatures abruptly becomes sharp upon entering the high-field phase,
indicating a dramatic change of the magnetic-flux properties. We attribute the
high-field phase to the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state, where the
formation of planar nodes gives rise to a segmentation of the flux-line
lattice. We point out that strongly orbital-dependent pairing as well as
spin-orbit interactions, the multiband nature, and the extremely small Fermi
energy are important for the formation of the FFLO state in FeSe.
