Radar-based personal identification and fall detection have received considerable attention in smart healthcare scenarios. Deep learning algorithms have been introduced to improve the performance of non-contact radar sensing applications. However, the original Transformer network is not suitable for multi-task radar-based applications to effectively extract temporal features from time-series radar signals. This article proposes the Multi-task Learning Radar Transformer (MLRT): a personal Identification and fall detection network based on IR-UWB radar. The proposed MLRT utilizes the attention mechanism of Transformer as its core to automatically extract features for personal identification and fall detection from radar time-series signals. Multi-task learning is applied to exploit the correlation between the personal identification task and the fall detection task, enhancing the performance of discrimination for both tasks. In order to suppress the impact of noise and interference, a signal processing approach is employed including DC removal and bandpass filtering, followed by clutter suppression using a RA method and Kalman filter-based trajectory estimation. An indoor radar signal dataset is generated with 11 persons under one IR-UWB radar, and the performance of MLRT is evaluated using this dataset. The measurement results show that the accuracy of MLRT improves by 8.5% and 3.6% for personal identification and fall detection, respectively, compared to state-of-the-art algorithms. The indoor radar signal dataset and the proposed MLRT source code are publicly available.