  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 Walsh
extended phase plane. The Banach space $X$ is assumed to be UMD and
sufficiently close to a Hilbert space in an interpolative sense. Our embedding
bounds imply $L^p$ bounds and sparse domination for the Banach-valued tritile
operator, a discrete model of the Banach-valued bilinear Hilbert transform.
