  We study the intrinsic anisotropy of the superconducting state in
KFe$_2$As$_2$, using small-angle neutron scattering to image the vortex lattice
as the applied magnetic field is rotated towards the FeAs crystalline planes.
The anisotropy is found to be strongly field dependent, indicating multiband
superconductivity. Furthermore, the high field anisotropy significantly exceeds
that of the upper critical field, providing further support for Pauli limiting
in KFe$_2$As$_2$ for field applied in the basal plane. The effect of Pauli
paramagnetism on the unpaired quasiparticles in the vortex cores is directly
evident from the ratio of scattered intensities due to the longitudinal and
transverse vortex lattice field modulation.
