The advent of accessible artificial intelligence (AI) tools and systems has begun a new era for creative expression, challenging us to gain a better understanding of human-AI collaboration and creativity. In this paper, we introduce Human-AI Songwriting Processes Dataset (HAISP), consisting of 34 coded submissions from the 2023 AI Song Contest teams. This dataset offers a resource for exploring the complex dynamics of AI-supported songwriting processes, facilitating investigations into the possibilities and challenges posed by AI in creative endeavors. Overall, HAISP contributes to advancing understanding of human-AI co-creation from the users' perspective. Furthermore, we outline potential use cases for the dataset, ranging from analyzing AI tools utilized in songwriting to gaining insights into users' ethical considerations and expanding creative possibilities. This can help to inform both scholarly inquiry and practical applications in music composition and beyond.