We provide necessary and sufficient conditions for a 1-jet (f,G):E→R×X to admit an extension (F,∇F) for some F∈C1,ω(X). Here E stands for an arbitrary subset of a Hilbert space X and ω is a modulus of continuity. As a corollary, in the particular case X=Rn, we obtain an extension (nonlinear) operator whose norm does not depend on the dimension n. Furthermore, we construct extensions (F,∇F) in such a way that: (1) the (nonlinear) operator (f,G)↦(F,∇F) is bounded with respect to a natural seminorm arising from the constants in the given condition for extension (and the bounds we obtain are almost sharp); (2) F is given by an explicit formula; (3) (F,∇F) depend continuously on the given data (f,G); (4) if f is bounded (resp. if G is bounded) then so is F (resp. F is Lipschitz). We also provide similar results on superreflexive Banach spaces.