Let \(\rho(\delta),\;0\leq \delta\leq 1\), be a modulus of smoothness and \(B\) a separable Banach space. Let \(H_{\rho}^{\circ}(B)\) be the space of all functions \(x:[0,1]\to B\) such that \(\| x(t+h)-x(t)\| _B=o(\rho(| h| ))\) uniformly in \(t\in [0,1)^d\) with the natural Hölder norm. The authors establish a sufficient condition for a random element \(\xi\) in \(H_{\rho}^{\circ}(B)\) to satisfy the central limit theorem, i.e., the convergence in distribution of \(n^{-1/2}\sum_1^n\xi_i\), where \(\xi_i\)'s are independent copies of \(\xi\). The conditions are formulated for the modulus \(\rho\), the space \(B\) and the second differences \(\Delta_h^2\xi(t):=\xi(t+h)+\xi(t-h)-2\xi(t)\) of the random element \(\xi\).