  We give an extension of Rubio de Francia's extrapolation theorem for
functions taking values in UMD Banach function spaces to the multilinear
limited range setting. In particular we show how boundedness of an
$m$-(sub)linear operator \[T:L^{p_1}(w_1^{p_1})\times\cdots\times
L^{p_m}(w_m^{p_m})\to L^p(w^p) \] for a certain class of Muckenhoupt weights
yields an extension of the operator to Bochner spaces $L^{p}(w^p;X)$ for a wide
class of Banach function spaces $X$, which includes certain Lebesgue, Lorentz
and Orlicz spaces.
  We apply the extrapolation result to various operators, which yields new
vector-valued bounds. Our examples include the bilinear Hilbert transform,
certain Fourier multipliers and various operators satisfying sparse domination
results.
