Data collected by IoT devices are often private and have a large diversity across users. Therefore, learning requires pre-training a model with available representative data samples, deploying the pre-trained model on IoT devices, and adapting the deployed model on the device with local data. Such an on-device adaption for deep learning empowered applications demands data and memory efficiency. However, existing gradient-based meta learning schemes fail to support memory-efficient adaptation. To this end, we propose p-Meta, a new meta learning method that enforces structure-wise partial parameter updates while ensuring fast generalization to unseen tasks. Evaluations on few-shot image classification and reinforcement learning tasks show that p-Meta not only improves the accuracy but also substantially reduces the peak dynamic memory by a factor of 2.5 on average compared to state-of-the-art few-shot adaptation methods.