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.
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