The Function-as-a-Service (FaaS) model is a great fit for data and event processing in the Internet of Things (IoT). Sending all data to a cloud-based FaaS platform, however, may cause performance and privacy issues. While these issues could be mitigated using edge computing, existing FaaS approaches, designed for the cloud, are too heavyweight to run on small, constrained edge nodes. In this paper, we propose tinyFaaS, a new FaaS system that is specifically designed for edge environments and their unique challenges. Our platform is lightweight enough to run on low-performance single machine edge nodes, provides a CoAP endpoint to support communication with low-power devices, and uses Docker containers to isolate tenants. We evaluate tinyFaaS through a proof-of-concept implementation that we benchmark and compare to state-of-the-art FaaS platforms. For IoT processing scenarios, we find that tinyFaaS outperforms existing systems by at least an order of magnitude.