A New York company called Ecovative Design grew mycelia into a custom drone-shaped chassis you see above. Unfortunately, some parts of the drone just can't be replaced with biodegradable materials for now, though the team tried to stay true to the idea and used silver nanoparticle ink (which can disintegrate along with the chassis) to print the device's circuits. For the test flight earlier this month, the team had to use propellers, controls and batteries taken from an ordinary quadcopter, but that might change in the future. You can read all about the development process on the scientists' website, where you can also download some 3D printable files of a few drone chassis concepts.
Update (November 26, 2014): Lynn Rothschild reached out to us in order to address some comments and clarify a few things. She wrote:
"This project was started as a way to minimize environmental impact when UAVs are used to monitor environmentally sensitive areas such as coral reefs. There are many other peaceful uses of UAVs where a biodegradable body would be useful. Do take a look at the team wiki, especially the "Human Practices" section. In terms of why we used each product, part of the choice was based on use and part on the rules of the iGEM competition, a primarily undergraduate synthetic biology competition. Cornstarch foam is an interesting idea, but since part of the idea is to grow it anywhere - including Mars - we would want to stay away from a corn-based product. While we were not aiming to create biodegradable electronics in a few month program with undergrads, obviously this is also of interest. Many thanks for your comments!"