Stable broadband microwave absorption is widely interested for their irreplaceable role in stealth technology for multiband microwave communication and detection. In this paper, a flexible broadband microwave absorber with self-recoverability was fabricated by self-assembling graphene sheets into polypropylene 3D framework. The as-fabricated absorber has a full-band absorption (more than 90%) in the working ranges of microwave communication and detection in military radars (2–40 GHz and 75–110 GHz), exhibiting larger bandwidth than the absorbers reported in literature. Furthermore, its bandwidth with more than 97% absorption (Reflection loss ≤ −15 dB) is as large as 62.73 GHz. The impressive absorption ability attributes to the impedance matching to free-space, coupling of multiple resonances and edge scattering of the microwave absorber, which could be manipulated by the pattern design of SA. For both the transverse electric and transverse magnetic polarizations, the broadband absorptions keep stable in a wide range of incidence angle, and they are recoverable after press and folding. The stable broadband absorption and recoverability make the flexible absorber suitable for the stealth of deformable objects like robots or foldable wings.