  In this paper, we formulate a "Grassmann extension" scheme for constructing
noncommutative (Grassmann) extensions of Yang-Baxter maps together with their
associated systems of P$\Delta$Es, based on the ideas presented in
\cite{Sokor-Kouloukas}. Using this scheme, we first construct a Grassmann
extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq
system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel,
integrable, Grassmann lattice Boussinesq system, and we derive its
$3D$-consistent limit. We show that some systems retain their $3D$-consistency
property in their Grassmann extension.
