Slow software deployment is an important issue in environments such as fog computing where this operation lies in the critical path of providing online services to the end users. The problem is even worse when the virtualized resources are made of modest machines such as single-board computers. This paper leverages the observation that, although Docker images are often very large, only a small fraction of their content is actually accessed by the containers during startup. We therefore propose to reorganize container images and download only the strictly necessary files before starting a container. The remaining image contents can then be downloaded asynchronously while the container is already running. Our performance evaluations show that FogDocker reduces container deployment times in the order of 3-5x on single-board computers and 2-3x on powerful servers, while incurring low runtime overhead and maintaining correctness even in the case a container accesses a file which was not downloaded yet.