Suppose we have a Fredholm section S, (the differential is Fredholm at the 0-set) of some
Banach vector bundle over X, transverse to the 0-section, with Fredholm index 1 and such that the 0-set of this section is a circle which is contractible in X, can S be pushed of the 0-section
through Fredholm sections? Note when the bundle is finite dimensional
this is easily seen to be the case. 

edit: as stated this is not right. Mike Usher pointed out that in finite dimensional case (say a rank k real vector bundle over X^{k+1}) there are secondary obstructions to having a non-vanishing section. Since these obstructions have to do with $\pi_k (S^{k-1})$ it is not even obvious how  to extend this to Fredholm setting. 
I guess the right question is then can one formulate obstruction theory in the Fredholm setting?