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We propose in this paper a Quantized Distilled Low-Rank Neural Radiance Field (QDLR-NeRF) representation for the task of light field compression. While existing compression methods encode the set of light field sub-aperture images, our proposed method learns an implicit scene representation in the form of a Neural Radiance Field (NeRF), which also enables view synthesis. To reduce its size, the model is first learned under a Low-Rank (LR) constraint using a Tensor Train (TT) decomposition within an Alternating Direction Method of Multipliers (ADMM) optimization framework. To further reduce the model's size, the components of the tensor train decomposition need to be quantized. However, simultaneously considering the optimization of the NeRF model with both the low-rank constraint and rate-constrained weight quantization is challenging. To address this difficulty, we introduce a network distillation operation that separates the low-rank approximation and the weight quantization during network training. The information from the initial LR-constrained NeRF (LR-NeRF) is distilled into a model of much smaller dimension (DLR-NeRF) based on the TT decomposition of the LR-NeRF. We then learn an optimized global codebook to quantize all TT components, producing the final QDLR-NeRF. Experimental results show that our proposed method yields better compression efficiency compared to state-of-the-art methods, and it additionally has the advantage of allowing the synthesis of any light field view with high quality.