Why do we adopt new rules, such as social distancing? Although human sciences research stresses the key role of social influence in behaviour change, most COVID‐19 campaigns emphasize the disease’s medical threat. In a global data set (n = 6,674), we investigated how social influences predict people’s adherence to distancing rules during the pandemic. Bayesian regression analyses controlling for stringency of local measures showed that people distanced most when they thought their close social circle did. Such social influence mattered more than people thinking distancing was the right thing to do. People’s adherence also aligned with their fellow citizens, but only if they felt deeply bonded with their country. Self‐vulnerability to the disease predicted distancing more for people with larger social circles. Collective efficacy and collectivism also significantly predicted distancing. To achieve behavioural change during crises, policymakers must emphasize shared values and harness the social influence of close friends and family.