It is well known that the Rank Theorem for $C^1$ maps can be obtained as a consequence of the Implicit Mapping Theorem and the Inverse Mapping Theorem for $C^1$ maps. See for example Zorich vol 1.
It is also well known that the Implicit and Inverse Mapping theorems are a consequence of the Banach's Fixed Point Theorem. See Pugh for a proof of Implicit Mapping Theorem directly from Banach's Fixed Point Theorem. See Rudin for a proof of the Inverse Mapping Theorem directly from Banach's Fixed Point Theorem.
Obviously, for what was exposed in the previous paragraph, a proof (or several proofs) of Rank Theorem for $C^1$ mapping by means of the Banach Fixed Point Theorem exists. In other words, the affirmations of the Banach fixed point theorem and Rank Theorem for $C^1$ mapping are not independent.
In view of these considerations, my question is as follows. Is there any textbook with a proof of the rank theorem using Banach's Fixed Point Theorem?
In the absence of a textbook, I would be very grateful for articles and lecture notes.