  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.
