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.

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.)

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)