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The celebrated Gromov's non-squeezing theorem of 1985 says that the unit ball $B^n$ in $C^n$ can be symplectically embedded in the "cylinder" $rB^1 \times C^{n-1}$ of radius r only if $r\ge 1$. Hamiltonian differential equations provide examples of symplectic transformations in infinite dimension. Known results on the non-squeezing property in Hilbert spaces cover compact perturbations of linear symplectic transformations and several specific non-linear PDEs, including the periodic Korteweg - de Vries equation and the periodic cubic Schr\"odinger equation. We prove a new version of the non-squeezing theorem for Hilbert spaces. We apply the result to the discrete nonlinear Schr\"odinger equation. This work is joint with Alexander Sukhov.