Bacterial biofilms consist of a complex network of biopolymers embedded with microorganisms, and together these components form a physically robust structure that enables bacteria to grow in a protected environment. This structure can help unwanted biofilms persist in situations ranging from chronic infection to the biofouling of industrial equipment, but under certain circumstances it can allow the biofilm to disperse and colonize new niches. Mechanical properties are therefore a key aspect of biofilm life. In light of the recently discovered growth-induced compressive stress present within a biofilm, we studied the mechanical behavior of Bacillus subtilis pellicles, or biofilms at the air-liquid interface, and tracked simultaneously the force response and macroscopic structural changes during elongational deformations. We observed that pellicles behaved viscoelastically in response to small deformations, such that the growth-induced compressive stress was still present, and viscoplastically at large deformations, when the pellicles were under tension. In addition, by using particle imaging velocimetry we found that the pellicle deformations were nonaffine, indicating heterogeneous mechanical properties with the pellicle being more pliable near attachment surfaces. Overall, our results indicate that we must consider not only the viscoelastic but also the viscoplastic and mechanically heterogeneous nature of these structures to understand biofilm dispersal and removal.