In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls $$\mathbb{B}$$ B and $$\mathbb{B'}$$ B ′ in separable complex Hilbert spaces $$\mathcal{H}$$ H and $$\mathcal{H'}$$ H ′ , respectively. It is found that if the mapping f ∈ C1+α at $${z_0} \in \partial \mathbb{B}$$ z 0 ∈ ∂ B with $$f\left( {{z_0}} \right) = {w_0} \in \partial \mathbb{B}'$$ f ( z 0 ) = w 0 ∈ ∂ B ′ , then the Fréchet derivative operator Df(z0) maps the tangent space $${T_{{z_0}}}(\partial {\mathbb{B}^n})$$ T z 0 ( ∂ B n ) to $${T_{{w_0}}}(\partial {\mathbb{B}'})$$ T w 0 ( ∂ B ′ ) , the holomorphic tangent space $$T_{{z_0}}^{(1,0)}(\partial {\mathbb{B}^n})$$ T z 0 ( 1 , 0 ) ( ∂ B n ) to $$T_{{w_0}}^{(1,0)}(\partial {\mathbb{B}'})$$ T w 0 ( 1 , 0 ) ( ∂ B ′ ) , respectively.