Light field (LF) imaging enables new possibilities for digital imaging, such as digital refocusing, changing of focus plane, changing of viewpoint, scene-depth estimation, and 3D scene reconstruction, by capturing both spatial and angular information of light rays. However, one main problem in dealing with LF data is its sheer volume. In this context, efficient compression methods are needed for such a particular type of content. In this paper, we propose a content-based LF image-compression method with Gaussian process regression to improve the compression efficiency and accelerate the prediction procedure. First, the LF image is fed to the intra-frame codec of HEVC. In the prediction procedure, the prediction units (PUs) are classified as non-homogenous texture units, homogenous texture units, and visually flat units, based on the content property of the LF image. For each category, we design a corresponding Gaussian process regression (GPR)-based prediction method. Moreover, we propose a classification mechanism to exactly decide to which category the current PU belongs, so as to adjust the trade-off between the computational burden and the LF image coding efficiency. Experimental results demonstrate that the proposed LF image compression method is superior to several other state-of-the-art compression methods in terms of different quality metrics. Furthermore, the proposed method can also achieve a good visual quality of views rendered from decoded LF contents.