The solution by Raynaud and Harbater of Abyhankar’s conjecture concerning fundamental groups of curves in positive characteristic uses the rigid analytic GAGA theorems (whose proofs are very similar to Serre’s proofs in the complex-analytic case).
Work of Kisin on modularity of Galois representations makes creative use of rigid-analytic spaces associated to Galois deformation rings.
An original article is
John Tate, Rigid analytic spaces, Invent. Math. 12:257–289, 1971.
and for the construction of the generic fiber of formal schemes over the ring of integers of
Michel Raynaud, Géométrie analytique rigide d’après Tate, Kiehl,⋯, Table Ronde d’Analyse non archimédienne (Paris, 1972), pp. 319–327. Bull. Soc. Math. France, Mem. No. 39–40, Soc. Math. France, Paris, 1974, MR470254
Introductions are in
Johannes Nicaise, Formal and rigid geometry: an intuitive introduction, and some applications (pdf)