Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) and which includes proofs of the usual statements in geometric topology like the existence of collars or isotopy extension theorems in the generality of manifolds with corners?

Most of the common textbooks treat the case without corners nor boundary and mention the case of boundaries. Some of them take care of boundaries more closely, but I am not aware of a detailed reference covering the situation with corners.

Differential Topology, by Margalef-Roif and Dominguez, builds the standard smooth manifold theory (inverse function theorem, submanifolds, transversality, etc) in the level of generality of Banach manifolds with corners.

I don't actually know a source that deals with isotopy extension and collar neighborhoods. Probably your best bet is just to carefully check the details of what happens in that setting yourself.