We study the space of locally L-analytic functions on the ring of integers in L, where L is a finite extension of Qp. We show that the dual of this space is a ring isomorphic to the ring of rigid functions on a certain rigid variety X. We show that the variety X is isomorphic to the open unit disk over Cp, but not over any discretely valued extension field of L; it is a "twisted form" of the open unit disk. In the ring of functions on X, the classes of closed, finitely generated, and invertible ideals coincide, but unless L=Qp not all finitely generated ideals are principal.
The paper uses Lubin-Tate theory and results on p-adic Hodge theory. We give several applications, including one to the construction of p-adic L-functions for supersingular elliptic curves.