The Object-Based Image Coding (OBIC) that was extensively studied about two decades ago, promised a vast application perspective for both ultra-low bitrate communication and high-level semantical content understanding, but it had rarely been used due to the inefficient compact representation of object with arbitrary shape. A fundamental issue behind is how to efficiently process the arbitrary-shaped objects at a fine granularity (e.g., feature element or pixel wise). To attack this, we have proposed to apply the element-wise masking and compression by devising an object segmentation network for image layer decomposition, and parallel convolution-based neural image compression networks to process masked foreground objects and background scene separately. All components are optimized in an end-to-end learning framework to intelligently weigh their (e.g., object and background) contributions for visually pleasant reconstruction. We have conducted comprehensive experiments to evaluate the performance on PASCAL VOC dataset at a very low bitrate scenario (e.g., $lesssim$0.1 bits per pixel - bpp) which have demonstrated noticeable subjective quality improvement compared with JPEG2K, HEVC-based BPG and another learned image compression method. All relevant materials are made publicly accessible at https://njuvision.github.io/Neural-Object-Coding/.