Time-lapse microscopy has long been used to record cell lineage trees. Successful construction of a lineage tree requires tracking and preserving the identity of multiple cells across many images. If a single cell is misidentified the identity of all its progeny will be corrupted and inferences about heritability may be incorrect. Successfully avoiding such identity errors is challenging, however, when studying highly-motile cells such as T lymphocytes which readily change shape from one image to the next. To address this problem, we developed DeepKymoTracker, a pipeline for combined tracking and segmentation. Central to DeepKymoTracker is the use of a seed, a marker for each cell which transmits information about cell position and identity between sets of images during tracking, as well as between tracking and segmentation steps. The seed allows a 3D convolutional neural network (CNN) to detect and associate cells across several consecutive images in an integrated way, reducing the risk of a single poor image corrupting cell identity. DeepKymoTracker was trained extensively on synthetic and experimental T lymphocyte images. It was benchmarked against five publicly available, automatic analysis tools and outperformed them in almost all respects. The software is written in pure Python and is freely available. We suggest this tool is particularly suited to the tracking of cells in suspension, whose fast motion makes lineage assembly particularly difficult.