Dockerfile flakiness—unpredictable temporal build failures caused by external dependencies and evolving environments—undermines deployment reliability and increases debugging overhead. Unlike traditional Dockerfile issues, flakiness occurs without modifications to the Dockerfile itself, complicating its resolution. In this work, we present the first comprehensive study of Dockerfile flakiness, featuring a nine-month analysis of 8,132 Dockerized projects, revealing that around 10% exhibit flaky behavior. We propose a taxonomy categorizing common flakiness causes, including dependency errors and server connectivity issues. Existing tools fail to effectively address these challenges due to their reliance on pre-defined rules and limited generalizability. To overcome these limitations, we introduce FLAKIDOCK, a novel repair framework combining static and dynamic analysis, similarity retrieval, and an iterative feedback loop powered by Large Language Models (LLMs). Our evaluation demonstrates that FLAKIDOCK achieves a repair accuracy of 73.55%, significantly surpassing state-of-the-art tools and baselines.