Underwater mapping with mobile robots has a wide range of applications, and good models are lacking for key parts of the problem, such as sensor behavior. The specific focus here is the huge environmental problem of underwater litter, in the context of the Horizon 2020 SeaClear project, where a team of robots is being developed to map and collect such litter. No reinforcement-learning solution to underwater mapping has been proposed thus far, even though the framework is well suited for robot control in unknown settings. As a key contribution, this paper therefore makes a first attempt to apply deep reinforcement learning (DRL) to this problem by exploiting two state-of-the-art algorithms and making a number of mapping-specific improvements. Since DRL often requires millions of samples to work, a fast simulator is required, and another key contribution is to develop such a simulator from scratch for mapping seafloor objects with an underwater vehicle possessing a sonar-like sensor. Extensive numerical experiments on a range of algorithm variants show that the best DRL method collects litter significantly faster than a baseline lawn mower trajectory.