One interesting result in this circle
of ideas is p-adic uniformization, and it is described in many
places. In short, the main result is that covering spaces of certain
varieties exist as rigid analytic spaces, and that the canonical map
from the universal cover respects the galois action.
For a quick statement of this theorem in the context of Shimura Curves see Theorem 4.7 in "Heegner points, p-adic L-functions and the Cerednik-Drinfeld uniformization" (by M. Bertolini and H. Darmon) available here on their website. The introduction of their paper also contains a neat application.
Contact Information: psuchand -at- umich .dot. edu