Local sections of arithmetic fundamental groups of p-adic curves

We investigate sections of the arithmetic fundamental group pi_1(X) where X is either a smooth affinoid p-adic curve, or a formal germ of a p-adic curve, and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y, and the exact sequence of pi_1(X) splits, then index (Y)=1. We also exhibit a necessary and sufficient condition for a section of pi_1(X) to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite.