  We demonstrate strong coupling between travelling magnons in an Yttrium Iron
Garnet film and 3D microwave cavity photons at milli-Kelvin temperatures. The
coupling strength of $350$MHz or $7.3$\% of resonance frequency is observed.
The magnonic subsystem is represented by the Damon-Eshbach magnetostatic
surface wave with a distribution of wave numbers giving the linewidth of 15MHz.
The ways to improve this parameter are discussed. The energy gap in the
spectrum given by the Zeeman energy and the shape-anisotropy energy in the film
geometry give rise to a significant asymmetry of the double peak structure of
the photon-magnon avoided level crossing. A structure of two parallel YIG films
is investigated using the same re-entrant magnetostatic surface wave transducer
revealing a higher order magnon modes existing in both films. Combination of a
multi-post re-entrant cavity and multiple films is a potential base for
engineering both magnon and photon spectra.
