  In this paper, we construct a Grassmann extension of a Yang-Baxter map which
first appeared in [16] and can be considered as a lift of the discrete
potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension
satisfies the Yang-Baxter equation, and it admits a $3 \times 3$ Lax matrix.
Moreover, we show that it can be squeezed down to a system of lattice equations
which possesses a Lax representation and whose bosonic limit is the dpKdV
equation. Finally, we consider commutative analogues of the constructed
Yang-Baxter map and its associated quad-graph system, and we discuss their
integrability.
