Current state-of-the-art (SOTA) codec-based audio synthesis systems can mimic anyone's voice with just a 3-second sample from that specific unseen speaker. Unfortunately, malicious attackers may exploit these technologies, causing misuse and security issues. Anti-spoofing models have been developed to detect fake speech. However, the open question of whether current SOTA anti-spoofing models can effectively counter deepfake audios from codec-based speech synthesis systems remains unanswered. In this paper, we curate an extensive collection of contemporary SOTA codec models, employing them to re-create synthesized speech. This endeavor leads to the creation of CodecFake, the first codec-based deepfake audio dataset. Additionally, we verify that anti-spoofing models trained on commonly used datasets cannot detect synthesized speech from current codec-based speech generation systems. The proposed CodecFake dataset empowers these models to counter this challenge effectively.