Smartphone addiction, its association with smartphone use, and its predictors have not yet been studied in a European sample. This study investigated indicators of smartphone use, smartphone addiction, and their associations with demographic and health behaviour-related variables in young people. A convenience sample of 1,519 students from 127 Swiss vocational school classes participated in a survey assessing demographic and health-related characteristics as well as indicators of smartphone use and addiction. Smartphone addiction was assessed using a short version of the Smartphone Addiction Scale for Adolescents (SAS-SV). Logistic regression analyses were conducted to investigate demographic and health-related predictors of smartphone addiction. Smartphone addiction occurred in 256 (16.9%) of the 1,519 students. Longer duration of smartphone use on a typical day, a shorter time period until first smartphone use in the morning, and reporting that social networking was the most personally relevant smartphone function were associated with smartphone addiction. Smartphone addiction was more prevalent in younger adolescents (15–16 years) compared with young adults (19 years and older), students with both parents born outside Switzerland, persons reporting lower physical activity, and those reporting higher stress. Alcohol and tobacco consumption were unrelated to smartphone addiction. Different indicators of smartphone use are associated with smartphone addiction and subgroups of young people have a higher prevalence of smartphone addiction. The study provides the first insights into smartphone use, smartphone addiction, and predictors of smartphone addiction in young people from a European country, which should be extended in further studies.