Amidst global climate challenges, carbon emission trading has become the most important market-based environmental policy tool, attracting widespread attention for mitigating price volatility caused by extreme risks. This study applies the multivariate multi-quantile conditional autoregressive value-at-risk (MVMQ-CAVIaRX) model to measure extreme market risk and modifies the Diebold Yilmaz (DY) spillover index calculated using the time-varying parameter vector autoregressive model with exogenous variables (TVP-VARX) to examine the extreme risk structures and its time-varying characteristics of the European carbon emissions trading market. The relevant results are threefold. (1) Significant extreme risk spillover effects exist between the carbon market and the stock, commodity, exchange rate, and interest rate markets, influenced by economic risks and geopolitical risks. (2) In the average extreme risk structure of the carbon market, aside from itself, geopolitical risks contribute the most, followed by the stock and commodity markets, while the contributions of the exchange rate and interest rate are relatively small, with economic risks exerting a slow and steadily increasing influence on extreme risks in the carbon market over the forecast period. (3) The extreme risk structure of the carbon market exhibits significant time-varying characteristics, with contributions from related extreme market risks, geopolitical risks, and economic risks showing significant variations during important periods such as the COVID-19 pandemic and the Russia–Ukraine war. These findings have implications for carbon market policymakers to manage extreme risks.