  We prove that the class of trilinear multiplier forms with singularity over a
one dimensional subspace, including the bilinear Hilbert transform, admit
bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No
other assumption, for instance of Rademacher maximal function type, is made on
the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an
extension of the phase-space projection technique to the $\mathrm{UMD}$-valued
setting. This is then employed to obtain appropriate single tree estimates by
appealing to the $\mathrm{UMD}$-valued bound for bilinear Calder\'on-Zygmund
operators recently obtained by the same authors.
