  In this paper, we give a general boundary Schwarz lemma for holomorphic
mappings between unit balls in any dimensions. It is proved that if the mapping
$f\in C^{1+\alpha}$ at $z_0\in \partial \mathbb B^n$ with $f(z_0)=w_0\in
\partial \mathbb B^N$ for any $n,N\geq 1$, then the Jacobian matrix $J_f(z_0)$
maps the tangent space $T_{z_0}(\partial \mathbb B^n)$ to $T_{w_0}(\partial
\mathbb B^N)$, and the holomorphic tangent space $T^{(1,0)}_{z_0}(\partial
\mathbb B^n)$ to $T^{(1,0)}_{w_0}(\partial \mathbb B^N)$ as well.
