Recent advancements in 3D printing have revolutionized biomedical engineering by enabling the manufacture of complex and functional devices in a low-cost, customizable, and small-batch fabrication manner. Soft elastomers are particularly important for biomedical applications because they can provide similar mechanical properties as tissues with improved biocompatibility. However, there are very few biocompatible elastomers with 3D printability, and little is known about the material properties of biocompatible 3D printable elastomers. Here, we report a new framework to 3D print a soft, biocompatible, and biostable polycarbonate-based urethane silicone (PCU-Sil) with minimal defects. We systematically characterize the rheological and thermal properties of the material to guide the 3D printing process and have determined a range of processing conditions. Optimal printing parameters such as printing speed, temperature, and layer height are determined via parametric studies aimed at minimizing porosity while maximizing the geometric accuracy of the 3D-printed samples as evaluated via micro-CT. We also characterize the mechanical properties of the 3D-printed structures under quasistatic and cyclic loading, degradation behavior and biocompatibility. The 3D-printed materials show a Young's modulus of 6.9 ± 0.85 MPa and a failure strain of 457 ± 37.7% while exhibiting good cell viability. Finally, compliant and free-standing structures including a patient-specific heart model and a bifurcating arterial structure are printed to demonstrate the versatility of the 3D-printed material. We anticipate that the 3D printing framework presented in this work will open up new possibilities not only for PCU-Sil, but also for other soft, biocompatible and thermoplastic polymers in various biomedical applications requiring high flexibility and strength combined with biocompatibility, such as vascular implants, heart valves, and catheters.