3D face generation has achieved high visual quality and 3D consistency thanks to the development of neural radiance fields (NeRF). Recently, to generate and edit 3D faces with NeRF representation, some methods are proposed and achieve good results in decoupling geometry and texture. The latent codes of these generative models affect the whole face, and hence modifications to these codes cause the entire face to change. However, users usually edit a local region when editing faces and do not want other regions to be affected. Since changes to the latent code affect global generation results, these methods do not allow for fine-grained control of local facial regions. To improve local controllability in NeRF-based face editing, we propose LC-NeRF, which is composed of a Local Region Generators Module and a Spatial-Aware Fusion Module, allowing for local geometry and texture control of local facial regions. Qualitative and quantitative evaluations show that our method provides better local editing than state-of-the-art face editing methods. Our method also performs well in downstream tasks, such as text-driven facial image editing.