  In high temperature cuprate superconductors, the interlayer coupling between
the CuO$_2$ planes plays an important role in dictating superconductivity, as
indicated by the sensitive dependence of the critical temperature (T$_C$) on
the number of CuO$_2$ planes in one structural unit. In
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ superconductor with two CuO$_2$ planes in
one structural unit, the interaction between the two CuO$_2$ planes gives rise
to band splitting into two Fermi surface sheets (bilayer splitting) that have
distinct superconducting gap. The iron based superconductors are composed of
stacking of the FeAs/FeSe layers; whether the interlayer coupling can cause
similar band splitting and its effect on superconductivity remain unclear. Here
we report high resolution laser-based angle-resolved photoemission spectroscopy
(ARPES) measurements on a newly discovered iron based superconductor,
KCa$_2$Fe$_4$As$_4$F$_2$ (T$_C$=33.5\,K) which consists of stacking FeAs blocks
with two FeAs layers separated by insulating Ca$_2$F$_2$ blocks. Bilayer
splitting effect is observed for the first time that gives rise to totally five
hole-like Fermi surface sheets around the Brilliouin zone center. Band
structure calculations reproduce the observed bilayer splitting by identifying
interlayer interorbital interaction between the two FeAs layers within one FeAs
block. All the hole-like pockets around the zone center exhibit Fermi
surface-dependent and nodeless superconducting gap. The gap functions with
short-range antiferromagetic fluctuations are proposed and the gap symmetry can
be well understood when the interlayer pairing is considered. The particularly
strong interlayer pairing is observed for one of the bands. Our observations
provide key information on the interlayer coupling and interlayer pairing in
understanding superconductivity in iron based superconductors.
