In the sense of space and quantity a space is, quite generally, a presheaf on some category of test objects.
An $\infty$-space is some ∞-categorification of this.
With some category $Sp \subset PSh(C)$ of spaces (1-spaces, that is) fixed, an $\infty$-space may for instance be modeled as an ∞-groupoid internal to $Sp$.
If $Sp$ is for instance a category of smooth test objects, such as Diff or CartSp, then we have
a smooth 1-space is a smooth space;
a smooth 2-space is a Lie groupoid;
a smooth $\infty$-space is a Lie ∞-groupoid or smooth ∞-stack.
Generally, the notion of ∞-groupoid internal to $PSh(C)$ is naturally interpreted as a simplicial presheaf. Indeed, when equipped with the model structure on simplicial presheaves, simplicial presheaves are good models for ∞-stack (∞,1)-toposes.
The $\infty$-stack terminology is possibly more familiar than that of $\infty$-spaces. To some extent the usefulness of both terminologies depends on whether $Sp$ is a petit topos of presheaves of open subsets in some fixed topological space or a gros topos of presheaves on general test spaces, such as on (an essentially small) version of Top itself.
In the former case of a petit topos, the $\infty$-stack terminology may be more suggestive. In the latter case of a gros topos the $\infty$-space perspective typically conveys the right intuition for the objects under consideration much better.
The idea, that “space” is fundamentally to be interpreted in the sense of space and quantity is only partially compatible with wide-spread use of the word “space” to mean concretely topological space.
In fact, in the context of ”$\infty$-space” there is a subtle confusion of terms (possible) here, which deserves to be carefully sorted out:
In higher category theory topological spaces $X$ are often, via the homotopy hypothesis, effectively identified with their fundamental ∞-groupoids $\Pi(X)$. The ∞-groupoid $\Pi(X)$ – as an $\infty$-groupoid internal to Set – is really a discrete $\infty$-space in the above sense, namely an $\infty$-groupoid in the category of presheaves on the point.
Following the duality of space and quantity, the concept dual to $\infty$-space is ∞-quantity. See there for more details.
One place where the conceptual usefulness of interpreting categories internal to smooth spaces as smooth 2-spaces (and de-emphasizing conceptually their realization as simplicial sheaves) has been particularly amplified is the work