  We extend, in the free probability framework, an invariance principle for
multilinear homogeneous sums with low influences recently established in [E.
Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions
with low influences: invariance and optimality. {\it Ann. Math.} {\bf 171}, no.
1, 295-341]. To do so, a hypercontractivity property for those homogeneous sums
is necessary, and to prove it has turned out to be our main task. Finally, we
deduce from our extension several universality phenomenons, in the spirit of
the paper [I. Nourdin, G. Peccati and G. Reinert (2010). Invariance principles
for homogeneous sums: universality of Gaussian Wiener chaos. {\it Ann. Probab.}
{\bf 38}, no. 5, 1947-1985].
