Liquid Argon Time Projection Chambers (LArTPCs) are high resolution particle imaging detectors, employed by accelerator-based neutrino oscillation experiments for high precision physics measurements. While images of particle trajectories are intuitive to analyze for physicists, the development of a high quality, automated data reconstruction chain remains challenging. One of the most critical reconstruction steps is particle clustering: the task of grouping 3D image pixels into different particle instances that share the same particle type. In this paper, we propose the first scalable deep learning algorithm for particle clustering in LArTPC data using sparse convolutional neural networks (SCNN). Building on previous works on SCNNs and proposal free instance segmentation, we build an end-to-end trainable instance segmentation network that learns an embedding of the image pixels to perform point cloud clustering in a transformed space. We benchmark the performance of our algorithm on PILArNet, a public 3D particle imaging dataset, with respect to common clustering evaluation metrics. 3D pixels were successfully clustered into individual particle trajectories with 90% of them having an adjusted Rand index score greater than 92% with a mean pixel clustering efficiency and purity above 96%. This work contributes to the development of an end-to-end optimizable full data reconstruction chain for LArTPCs, in particular pixel-based 3D imaging detectors including the near detector of the Deep Underground Neutrino Experiment. Our algorithm is made available in the open access repository, and we share our Singularity software container, which can be used to reproduce our work on the dataset.