Biological invasions strongly increased during the last centuries and are challenging environmental managers worldwide. In this context, public acceptance of management measures is a key factor determining the long-term success of the control of invasive species. However, in the case of charismatic and iconic invasive species, the public has often been unwilling to accept strict management measures. Here, we studied the public perception of the Chinese windmill palm (Trachycarpus fortunei) in Switzerland, which is declared as invasive in southern Switzerland but also recognized as iconic. We conducted a nation-wide online survey in the multilingual and multicultural context of Switzerland, investigating the influence of social and cultural factors on the knowledge of, the attitude toward, and the willingness to control the invasive T. fortunei. Results confirm that the knowledge and perception of invasive plants have a strong social and cultural component and may vary greatly as a function of the cultural background, education level, age, and other social characteristics. Furthermore, information on the invasiveness of the focal species provided during the survey significantly affected informants’ perceptions, which are closely related to the acceptance of possible management and control measures. This allows us to highlight the importance of a holistic approach that includes targeted public information when dealing with biological invasions, especially in the case of charismatic and iconic species. Based on the obtained results, we suggest avenues for refining management and control strategies of T. fortunei in Switzerland, many of which generally applicable to other cases of invasive species.