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The paper introduces a novel \(L_\infty \)-constrained compression method for depth maps. The proposed method performs depth segmentation and depth prediction in each segment, encoding the resulting information as a base layer. The depth residuals are modeled using a Two-Sided Geometric Distribution, and distortion and entropy models for the quantized residuals are derived based on such distributions. A set of optimal quantizers is determined to ensure a fix rate budget at a minimum \(L_\infty \) distortion. A fixed-rate \(L_\infty \) codec design performing context-based entropy coding of the quantized residuals is proposed, which is able to efficiently meet user constraints on rate or distortion. Additionally, a scalable \(L_\infty \) codec extension is proposed, which enables encoding the quantized residuals in a number of enhancement layers. The experimental results show that the proposed \(L_\infty \) coding approach substantially outperforms the \(L_\infty \) coding extension of the state-of-the-art CALIC method.