  A locally gauge-invariant theory of the spin response of a thin film of
$^3$He-$B$ film is given that describes fluctuation effects arising from the
coupled dynamics of the superconducting order parameter (the collective mode)
and in-gap Majorana surface states. In contrast to a mean-field calculation of
the spin response, which predicts a nonzero imaginary longitudinal spin
susceptibility at frequencies inside the bulk gap due to absorption from the
Majorana states, our gauge-invariant theory shows that this response is
strongly suppressed above the collective mode frequency and vanishes if
dipole-dipole interactions are neglected. In the presence of dipole-dipole
interactions, in sufficiently thin films, and at ultra-low temperatures, the
Majorana states lead to a distinctive magnetic-field- and temperature-dependent
damping of the collective mode, a feature that may be observable in
longitudinal NMR experiments.
