  In conventional BCS superconductors, the quantum condensation of
superconducting electron pairs is understood as a Fermi surface (FS)
instability, in which the low-energy electrons are paired by attractive
interactions. Whether this explanation is still valid in high-Tc
superconductors such as cuprates and iron-based superconductors remains an open
question. In particular, a fundamentally different picture of the electron
pairs, which are believed to be formed locally by repulsive interactions, may
prevail. Here we report a high-resolution angle-resolved photoemission
spectroscopy study on LiFe1-xCoxAs. We reveal a large and robust
superconducting (SC) gap on a band sinking below the Fermi energy upon Co
substitution. The observed FS-free SC order is also the largest over the
momentum space, which rules out a proximity effect origin and indicates that
the SC order parameter is not tied to the FS as a result of a FS instability.
