For Banach spaces X and Y, we consider bifurcation from the line of trivial solutions for the equation F(lambda, u) = 0, where F : R x X -> Y with F(lambda, 0) = 0 for all lambda is an element of R. The focus is on the situation where F(lambda, center dot) is only Hadamard differentiable at 0 and Lipschitz continuous on some open neighbourhood of 0, without requiring any Frechet differentiability. Applications of the results obtained here to some problems involving nonlinear elliptic equations on R-N, where Frechet differentiability is not available, are presented in some related papers, which shed light on the relevance of our hypotheses.