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We prove modulation invariant embedding bounds from Bochner spaces$L^p(\mathbb{W};X)$ on the Walsh group to outer-$L^p$ spaces on the Walshextended phase plane. The Banach space $X$ is assumed to be UMD andsufficiently close to a Hilbert space in an interpolative sense. Our embeddingbounds imply $L^p$ bounds and sparse domination for the Banach-valued tritileoperator, a discrete model of the Banach-valued bilinear Hilbert transform.

Banach-valued modulation invariant Carleson embeddings and outer-$L^p$ spaces: the Walsh case