  We find evidence that the newly discovered Fe-based superconductor
KCa$_2$Fe$_4$As$_4$F$_2$ ($T_c~=~33.36(7)$~K) displays multigap
superconductivity with line nodes. Transverse field muon spin rotation
($\mu$SR) measurements show that the temperature dependence of the superfluid
density does not have the expected behavior of a fully-gapped superconductor,
due to the lack of saturation at low temperatures. Moreover, the data cannot be
well fitted using either single band models or a multiband $s$-wave model, yet
are well described by two-gap models with line nodes on either one or both of
the gaps. Meanwhile the zero-field $\mu$SR results indicate a lack of time
reversal symmetry breaking in the superconducting state, but suggest the
presence of magnetic fluctuations. These results demonstrate a different route
for realizing nodal superconductivity in iron-based superconductors. Here the
gap structure is drastically altered upon replacing one of the spacer layers,
indicating the need to understand how the pairing state is tuned by changes of
the asymmetry between the pnictogens located either side of the Fe planes.
