  Conventional superconductivity follows Bardeen-Cooper-Schrieffer(BCS) theory
of electrons-pairing in momentum-space, while superfluidity is the
Bose-Einstein condensation(BEC) of atoms paired in real-space. These properties
of solid metals and ultra-cold gases, respectively, are connected by the
BCS-BEC crossover. Here we investigate the band dispersions in
FeTe$_{0.6}$Se$_{0.4}$($T_c$ = 14.5 K $\sim$ 1.2 meV) in an accessible range
below and above the Fermi level($E_F$) using ultra-high resolution laser
angle-resolved photoemission spectroscopy. We uncover an electron band lying
just 0.7 meV ($\sim$ 8 K) above $E_F$ at the $\Gamma$-point, which shows a
sharp superconducting coherence peak with gap formation below $T_c$. The
estimated superconducting gap $\Delta$ and Fermi energy $\epsilon_F$ indicate
composite superconductivity in an iron-based superconductor, consisting of
strong-coupling BEC in the electron band and weak-coupling BCS-like
superconductivity in the hole band. The study identifies the possible route to
BCS-BEC superconductivity.
