Background: The dynamic character of the COVID-19 pandemic and its social consequences caused several medical and societal issues and dilemmas. The aim of our qualitative research was to capture and analyze attitudes and beliefs of convalescents who experienced mild symptoms of COVID-19 in the first wave of the pandemic and decided to donate their plasma for therapeutic purposes. Material and Methods: The article presents results of qualitative research conducted on the basis of grounded theory (GT) methodology. Empirical material includes 10 in-depth interviews conducted with respondents who had mild or asymptomatic disease and, after recovery, voluntarily donated their plasma to the Regional Centre for Blood Donation and Blood Treatment (RCKiK). Data were collected in May and June 2020 in Poland. Qualitative analysis was focused on the experience of convalescents who entered the social role of a sick person in individual, social, and organizational dimensions. Results: The social role of the patient in the narratives of convalescents was related to three stages: (1) initiation to the role, (2) staying in the COVID-19 patient role, and (3) leaving the role. Research results enabled the distinction of three basic descriptive categories (“ontological uncertainty”, “the global and individual dimension”, and “being sick in the disease-infected environment”), which became epistemological framework for a detailed description of the roles played by an individual COVID-19 patient during the pandemic. Conclusions: The disease, despite its mild course, generated a number of non-medical issues, and the entire process of being ill was burdened with institutional and emotional struggles. The experience of mild COVID-19 is significantly modified by disease institutionalization. These results may contribute to a better understanding of the psychosocial dimension of COVID-19 and convalescents’ motivations for plasma donation.