We present a novel attack against the Combined Charging System, one of the most widely used DC rapid charging technologies for electric vehicles (EVs). Our attack, Brokenwire, interrupts necessary control communication between the vehicle and charger, causing charging sessions to abort. The attack requires only temporary physical proximity and can be conducted wirelessly from a distance, allowing individual vehicles or entire fleets to be disrupted stealthily and simultaneously. In addition, it can be mounted with off-the-shelf radio hardware and minimal technical knowledge. By exploiting CSMA/CA behavior, only a very weak signal needs to be induced into the victim to disrupt communication - exceeding the effectiveness of broadband noise jamming by three orders of magnitude. The exploited behavior is a required part of the HomePlug Green PHY, DIN 70121 & ISO 15118 standards and all known implementations exhibit it. We first study the attack in a controlled testbed and then demonstrate it against eight vehicles and 20 chargers in real deployments. We find the attack to be successful in the real world, at ranges up to 47 m, for a power budget of less than 1 W. We further show that the attack can work between the floors of a building (e.g., multi-story parking), through perimeter fences, and from `drive-by' attacks. We present a heuristic model to estimate the number of vehicles that can be attacked simultaneously for a given output power. Brokenwire has immediate implications for a substantial proportion of the around 12 million battery EVs on the roads worldwide - and profound effects on the new wave of electrification for vehicle fleets, both for private enterprise and crucial public services, as well as electric buses, trucks and small ships. As such, we conducted a disclosure to the industry and discussed a range of mitigation techniques that could be deployed to limit the impact.