Code models, such as CodeBERT and CodeT5, offer general-purpose representations of code and play a vital role in supporting downstream automated software engineering tasks. Most recently, code models were revealed to be vulnerable to backdoor attacks. A code model that is backdoor-attacked can behave normally on clean examples but will produce pre-defined malicious outputs on examples injected with triggers that activate the backdoors. Existing backdoor attacks on code models use unstealthy and easy-to-detect triggers. This paper aims to investigate the vulnerability of code models with stealthy backdoor attacks. To this end, we propose AFRAIDOOR (Adversarial Feature as Adaptive Backdoor). AFRAIDOOR achieves stealthiness by leveraging adversarial perturbations to inject adaptive triggers into different inputs. We evaluate AFRAIDOOR on three widely adopted code models (CodeBERT, PLBART and CodeT5) and two downstream tasks (code summarization and method name prediction). We find that around 85% of adaptive triggers in AFRAIDOOR bypass the detection in the defense process. By contrast, only less than 12% of the triggers from previous work bypass the defense. When the defense method is not applied, both AFRAIDOOR and baselines have almost perfect attack success rates. However, once a defense is applied, the success rates of baselines decrease dramatically to 10.47% and 12.06%, while the success rate of AFRAIDOOR are 77.05% and 92.98% on the two tasks. Our finding exposes security weaknesses in code models under stealthy backdoor attacks and shows that the state-of-the-art defense method cannot provide sufficient protection. We call for more research efforts in understanding security threats to code models and developing more effective countermeasures.