Recent buzzes around big data, data science and artificial intelligence portray a data-driven future for healthcare. As a response, Europe's key players have stimulated the use of big data technologies to make healthcare more efficient and effective. Critical Data Studies and Science and Technology Studies have developed many concepts to reflect on such overly positive narratives and conduct critical policy evaluations. In this study, we argue that there is also much to be learned from studying how professionals in the healthcare field affectively engage with this strong European narrative in concrete big data projects. We followed twelve hospital-based big data pilots in eight European countries and interviewed 145 professionals (including legal, governance and ethical experts, healthcare staff and data scientists) between 2018 and 2020. In this study, we introduce the metaphor of dreams to describe how professionals link the big data promises to their own frustrations, ideas, values and experiences with healthcare. Our research answers the question: how do professionals in concrete data-driven initiatives affectively engage with European Union's data hopes in their ‘dreams’ – and with what consequences? We describe the dreams of being seen, of timeliness, of connectedness and of being in control. Each of these dreams emphasizes certain aspects of the grand narrative of big data in Europe, makes particular assumptions and has different consequences. We argue that including attention to these dreams in our work could help shine an additional critical light on the big data developments and stimulate the development of responsible data-driven healthcare.