In Def 2.5.1 in " Derived Quot schemes" by Ciocan-Fontanine and Kapranov, we can find the notion of dg-manifolds.

In detail, let $X$ be a dg-scheme over $k$ ($k$ : an algebraic closed field of $ch(k) = 0$), then $X = (X^0, \mathcal{O}_X^{\bullet})$ is called a dg-manifold if $X^0$ is a smooth algebraic variety and $\mathcal{O}^{\bullet}_X$ is locally isomorphic (as graded sheaves ) to $\operatorname{Sym}_{\mathcal{O}_X^0}(\bigoplus_i Q^{-i})$ where the degree $-i$-part $Q^{-i}$ are locally free sheaf on $X^0$ of finite ranks.

(a) Can we consider an analogous definition in the case $X^0$ is infinite dimensional or $Q^{-i}$ is infinite rank ? (Edit: Any reference is also welcome.)

(b) Moreover, if we can define that notion, are there any conditions such that an infinite dg-manifold can be quasi-isomorphic to some dg-manifolds defined by Ciocan-Fontanine and Kapranov.

At least, I will treat the example whose classical part is finite dimensional and their tangent complexes have finite cohomology at each point. (Edit: I am also considering fixing constructions of derived quot schemes by them because that constructions are failed according to this paper)

Edit: Can we also extend (b) in dg-stacks ? In particular, I am interested in the case of dg-stacks constructed from dg Lie algebras (i.e, for any given $L := \oplus_{i \geq 0 }L_i$, the underlying stack is $\mathcal{X}= [L_1/ \text{exp}(L_0)]$ and the corresponding sheaf $\mathcal{O}_{\mathcal{X}}^\bullet$ of dg-algebras on $\mathcal{X}$ is a descent of the sheaf of dg-algebras on $L_1$. From this construction, $(\mathcal{X}, \mathcal{O}^{\bullet}_{\mathcal{X}})$ can be called a "infinity dimensional smooth dg manifold" when $\mathcal{X}$ is infinite dimensional or $\mathcal{O}_{\mathcal{X}}^\bullet$ is generated by infinite rank bundles as above. See, also Section 1 in this paper).

To be precise, when the the classical part of $(\mathcal{X}, \mathcal{O}^{\bullet}_{\mathcal{X}})$ is finite dimensional and has tangent complexes with finite dimensional cohomology at each point, is this quasi-isomorphic to a smooth dg-stack defined by Cioca-Fontanine and Kapranov in page 31 in "Derived Hilbert schemes"?