We prove a global bifurcation result for an abstract equation of the type Lx + \lambda h(\lambda,x) = 0 , where L: E \to F is a linear Fredholm operator of index zero between Banach spaces and h\colon \mathbb R \times E \to F is a C^{1} (not necessarily compact) map. We assume that L is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set \Sigma of nontrivial solutions of the above equation (i.e. solutions (\lambda,x) with \lambda \neq 0 ) such that the closure of \Sigma contains a trivial solution (0,\bar x) . This result extends previous ones in which the compactness of h was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors.