Generating set of a complex lattice contains a basis

Question

Let $\Gamma\subset \mathbb{C}$ be a lattice, i.e. a set of the form $\gamma_1\mathbb{Z}+\gamma_2\mathbb{Z}$ where the set of complex numbers $\{\gamma_1, \gamma_2\}$ is $\mathbb{R}$-linearly independent. Suppose $S\subset \Gamma$ is a finite generating set of $\Gamma$. Does there always exist a subset of $S$ that is a basis of $\Gamma$ (i.e. an $\mathbb{R}$-linearly independent generating set)?