  In this paper, we introduce the class of Cohen-Macaulay (=CM) dg
(=differential graded) modules over Gorenstein dg algebras and study their
basic properties. We show that the category of CM dg modules forms a Frobenius
extriangulated category, in the sense of Nakaoka and Palu, and it admits almost
split extensions. We also study representation-finite $d$-self-injective dg
algebras $A$ in detail. In particular, we classify the Auslander-Reiten (=AR)
quivers of CM $A$ for those $A$ in terms of $(-d)$-Calabi-Yau (=CY)
configurations, which are Riedtmann's configuration for the case $d=1$. For any
given $(-d)$-CY configuration $C$, we show there exists a $d$-self-injective dg
algebra $A$, such that the AR quiver of CM $A$ is given by $C$. For type
$A_{n}$, by using a bijection between $(-d)$-CY configurations and certain
purely combinatorial objects which we call maximal $d$-Brauer relations given
by Coelho Sim\~oes, we construct such $A$ through a Brauer tree dg algebra.
