We prove a global bifurcation result for an abstract equation of the type , where is a linear Fredholm operator of index zero between Banach spaces and is a (not necessarily compact) map. We assume that is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation (i.e. solutions with ) such that the closure of contains a trivial solution . This result extends previous ones in which the compactness of was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors.