Social media provides many opportunities to monitor and evaluate political phenomena such as referendums and elections. In this study, we propose a set of approaches to analyze long-running political events on social media with a real-world experiment: the debate about Brexit, i.e., the process through which the United Kingdom activated the option of leaving the European Union. We address the following research questions: Could Twitter-based stance classification be used to demonstrate public stance with respect to political events? What is the most efficient and comprehensive approach to measuring the impact of politicians on social media? Which of the polarized sides of the debate is more responsive to politician messages and the main issues of the Brexit process? What is the share of bot accounts in the Brexit discussion and which side are they for? By combining the user stance classification, topic discovery, sentiment analysis, and bot detection, we show that it is possible to obtain useful insights about political phenomena from social media data. We are able to detect relevant topics in the discussions, such as the demand for a new referendum, and to understand the position of social media users with respect to the different topics in the debate. Our comparative and temporal analysis of political accounts can detect the critical periods of the Brexit process and the impact they have on the debate.