Abstract: In 1900 Hilbert asked whether every locally Euclidean group is a Lie group. The question was answered affirmatively in 1950, by the joint effort of several mathematicians. Before that, it was proved in 1946 by Segal, that a one-sided differentiability condition is enough, for a group to be a Lie group. We show, that this can be generalized to groups modeled on Banach spaces. We also show, that for commutative groups, some Lipschitz condition on group operations is enough to get linear structure in the group. To do so, we introduce a notion of non-linear type in Banach spaces. The techniques give, as a byproduct, several results on Banach spaces.