  We report an unusual persistence of superconductivity against high magnetic
fields in the iron chalcogenide film FeTe:O$_{x}$ below ~ 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.
