False data injection attack (FDIA) is a critical security issue in power system state estimation. In recent years, machine learning (ML) techniques, especially deep neural networks (DNNs), have been proposed in the literature for FDIA detection. However, they have not considered the risk of adversarial attacks, which were shown to be threatening to DNN's reliability in different ML applications. In this paper, we evaluate the vulnerability of DNNs used for FDIA detection through adversarial attacks and study the defensive approaches. We analyze several representative adversarial defense mechanisms and demonstrate that they have intrinsic limitations in FDIA detection. We then design an adversarial-resilient DNN detection framework for FDIA by introducing random input padding in both the training and inference phases. Extensive simulations based on an IEEE standard power system show that our framework greatly reduces the effectiveness of adversarial attacks while having little impact on the detection performance of the DNNs.