We prove a global bifurcation result for an equation of the type L x + λ ( h ( x ) + k ( x ) ) = 0 , where L : E     →     F is a linear Fredholm operator of index zero between Banach spaces, and, given an open subset Ω of E , h , k : Ω × [ 0 , + ∞ )     →     F are C 1 and continuous, respectively. Under suitable conditions, we prove the existence of an unbounded connected set of nontrivial solutions of the above equation, that is, solutions ( x , λ ) with λ ≠ 0 , whose closure contains a trivial solution ( x ¯ , 0 ) . The proof is based on a degree theory for a special class of noncompact perturbations of Fredholm maps of index zero, called α -Fredholm maps, which has been recently developed by the authors in collaboration with M. Furi.