We report an unusual persistence of superconductivity against high magnetic fields in the iron-chalcogenide film FeTe:Ox below <$Esymbol Ы~2,5> K. Instead of saturating like a mean-field behavior with a single order parameter, the measured low-temperature upper critical field increases progressively, suggesting a large supply of superconducting states accessible via magnetic field or low-energy thermal fluctuations. We demonstrate that superconducting states of finite momenta can be realized within the conventional theory, despite its questionable applicability. Our findings reveal a fundamental characteristic of superconductivity and electronic structure in the strongly-correlated iron-based superconductors.

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