In order to move beyond mean-based connectedness measures in the cryptocurrency market and capture connectedness under extreme events, this paper applies quantile-based connectedness measures based on the variance decomposition of a quantile vector autoregression model. Based on the daily price data of seven leading cryptocurrencies from August 8, 2015 to December 31, 2020, the results show that the connectedness measures in the left and right tails are much higher than those in the mean and median of the conditional distribution. This suggests that return connectedness strengthens with shock size for both positive and negative shocks, indicating that return shocks propagate more intensely during extreme events relative to calm periods. While this result shows the instability of the system of connectedness under extreme events such as the COVID-19 outbreak, it implies the need to move beyond mean-based connectedness measures to understand the return connectedness under extreme negative and extreme positive shocks. Further analyses based on rolling windows show evidence of asymmetry between the behaviour of return spillovers in lower quantiles and upper quantiles. The results are robust to several alternative choices.