Conducting data analysis typically involves authoring code to transform, visualize, analyze, and interpret data. Large language models (LLMs) are now capable of generating such code for simple, routine analyses. LLMs promise to democratize data science by enabling those with limited programming expertise to conduct data analyses, including in scientific research, business, and policymaking. However, analysts in many real-world settings must often exercise fine-grained control over specific analysis steps, verify intermediate results explicitly, and iteratively refine their analytical approaches. Such tasks present barriers to building robust and reproducible analyses using LLMs alone or even in conjunction with existing authoring tools (e.g., computational notebooks). This paper introduces Flowco, a new mixed-initiative system to address these challenges. Flowco leverages a visual dataflow programming model and integrates LLMs into every phase of the authoring process. A user study suggests that Flowco supports analysts, particularly those with less programming experience, in quickly authoring, debugging, and refining data analyses.