  We prove $L^p$-bounds for the bilinear Hilbert transform acting on functions
valued in intermediate UMD spaces. Such bounds were previously unknown for UMD
spaces that are not Banach lattices. Our proof relies on bounds on embeddings
from Bochner spaces $L^p(\mathbb{R};X)$ into outer Lebesgue spaces on the
time-frequency-scale space $\mathbb{R}^3_+$.
