Solutions to first order equation with Clifford algebra elements

In the same way one can show that $\nabla^{2}\theta=0$ has only one smooth solution, namely $\theta=0$, I would like to show that

$$\gamma^{i}\partial_{i}\epsilon=0$$ has only one smooth solution, where $\gamma^{i}$ is a Dirac gamma matrix (or an element of the Clifford algebra), and $\epsilon$ is a spinorial quantity (which may or may not be relevant to finding a smooth solution, I am not sure).