Asteroids preserve important information about the origin and evolution of the Solar System. Three-dimensional (3D) surface models of asteroids are essential for exploration missions and scientific research. Regular methods for 3D surface reconstruction of asteroids, such as stereo-photogrammetry (SPG), usually struggle to reconstruct textureless areas and can only generate sparse surface models. Stereo-photoclinometry (SPC) can reconstruct pixel-wise topographic details but its performance depends on the availability of images obtained under different illumination conditions and suffers from uncertainties related to surface reflectance and albedo. This paper presents Asteroid-NeRF, a novel deep-learning method for 3D surface reconstruction of asteroids that is based on the state-of-the-art neural radiance field (NeRF) method. Asteroid-NeRF uses a signed distance field (SDF) to reconstruct a 3D surface model from multi-view posed images of an asteroid. In addition, Asteroid-NeRF incorporates appearance embedding to adapt to different illumination conditions and to maintain the geometric consistency of a reconstructed surface, allowing it to deal with the different solar angles and exposure conditions commonly seen in asteroid images. Moreover, Asteroid-NeRF incorporates multi-view photometric consistency to constrain the SDF, enabling optimised reconstruction. Experimental evaluations using actual images of asteroids Itokawa and Bennu demonstrate the promising performance of Asteroid-NeRF, complementing traditional methods such as SPG and SPC. Furthermore, due to the global consistency and pixel-wise training of Asteroid-NeRF, it produces highly detailed surface reconstructions. Asteroid-NeRF offers a new and effective solution for high-resolution 3D surface reconstruction of asteroids that will aid future exploratory missions and scientific research on asteroids.