We present a deep learning framework for volumetric speckle reduction in optical coherence tomography (OCT) based on a conditional generative adversarial network (cGAN) that leverages the volumetric nature of OCT data. In order to utilize the volumetric nature of OCT data, our network takes partial OCT volumes as input, resulting in artifact-free despeckled volumes that exhibit excellent speckle reduction and resolution preservation in all three dimensions. Furthermore, we address the ongoing challenge of generating ground truth data for supervised speckle suppression deep learning frameworks by using volumetric non-local means despeckling-TNode to generate training data. We show that, while TNode processing is computationally demanding, it serves as a convenient, accessible gold-standard source for training data; our cGAN replicates efficient suppression of speckle while preserving tissue structures with dimensions approaching the system resolution of non-local means despeckling while being two orders of magnitude faster than TNode. We demonstrate fast, effective, and high-quality despeckling of the proposed network in different tissue types acquired with three different OCT systems compared to existing deep learning methods. The open-source nature of our work facilitates re-training and deployment in any OCT system with an all-software implementation, working around the challenge of generating high-quality, speckle-free training data.