  The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle
modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be
adapted over general regular base schemes. However, there are simple examples
to show that Gersten complex fails to be exact for cycle modules in general if
the base is not a field. The goal of this article is to show that, for a
restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying
an extra axiom, called here as R5, the expected properties of exactness of
Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R5 is vacuously
satisfied when the base is a perfect field and it is also satisfied by $K^{MW}$
over any base. As a corollary, we obtain the strict $\mathbb{A}^1$-invariance
and the exactness of Gersten complex for $K^{MW}$ over an excellent DVR.
