Warning: In this answer, I'll comment about things a lot of MSE users know much better than I. I hope they'll correct and/or complete this answer.
In André Weil's book Basic Number Theory.
Wedderburn's Little Theorem is used in an essential way to compute the Brauer group of a local field.
The use of WLT is at the same time conspicuous and hidden.
It is conspicuous because WLT is Theorem 1 of Chapter 1, and is stated on top of page 1.
It is hidden because WLT is never (as far as I can see) referred to explicitly in the sequel.
But if one looks closely, one sees that it is implicitly used a lot of times.
The first time is in Corollary 1 to Theorem 2 page 2.
[In this post, division rings will be called "fields", that is "not necessarily commutative fields", to stick to Weil's terminology.]
WLT is tacitly used to conclude that the residue field of a (not necessarily commutative) non-archimedian local field is a finite commutative field.
This enables Weil to describe in a very precise way the structure of such a (not necessarily commutative) non-archimedian local field, viewed as a division algebra over its center: see Proposition 5 page 20.
This Proposition is the culminating point of Chapter 1, and then it is used in a crucial way in:
the comment following Definition 6 page 184,
the proof of Theorem 1 page 222,
the proof of Corollary 1 page 223.
A particularly important excerpt is the paragraph just before Theorem 2 page 224. Here is a part of this paragraph:
As we identify the Brauer group $B(K)$ with the group $H(K)$ considered in theorem 1 and its corollaries, we may consider the mapping $\eta$ defined in corollary 2 of th. 1 as an isomorphism of $B(K)$ onto the group of roots of $1$ in $\mathbb C$; ...