Abstract: Cancer remains a significant global health challenge, and particle therapy using protons and heavy ions (such as carbon) represents an advanced modality with superior dose conformality via the Bragg peak. However, the widespread application of carbon ion therapy is severely hindered by the large physical footprint and high construction costs associated with conventional normal-conducting synchrotrons. The work aims to develop a highly compact superconducting synchrotron capable of accelerating multiple ion species, including carbon, helium, and oxygen, thereby reducing the facility footprint and improving the accessibility of this therapy. The target maximum energy is 430 MeV/u for carbon ions, corresponding to a maximum magnetic rigidity of 6.6 Tm. To achieve exceptional compactness, the proposed design utilizes strongly curved canted-cosine-theta (CCT) superconducting magnets, which integrate both bending and strong focusing capabilities. The accelerator lattice, multi-turn injection process, and third-order resonance slow extraction scheme were designed and optimized using the MAD-X code and the pyOrbit code. The designed synchrotron has a two-fold symmetric racetrack layout consisting of four 90° bending sections, achieving a remarkably small circumference of approximately 32.65 meters. Because the proposed CCT magnets are strongly curved with a small bending radius of 2 meters, they inherently generate complex multipole components and fringe fields that are distinctly different from those found in standard straight magnets. Therefore, a comprehensive 3D magnetic field map was generated using the Opera simulation software, and the magnetic field quality was analyzed using a Taylor series expansion method. Furthermore, to accurately evaluate the combined effects of these non-ideal high-order magnetic fields on beam behavior, single-particle and multi-particle tracking simulations were conducted. The lattice design successfully maintains low horizontal beta functions inside the CCT magnets, significantly minimizing the required physical aperture and associated fabrication costs. Injection simulations indicate that the multi-turn injection method achieves a high gain factor of 25.9 over 500 turns, easily satisfying the clinical beam intensity requirements for patient treatment. The slow extraction scheme provides a sufficient physical separation gap at the magnetic septum while safely guiding the extracted beam through downstream superconducting magnets. Crucially, the beam dynamics simulations reveal that despite the presence of nonlinear components such as sextupole, octupole, and decapole fields intrinsically generated by the curved CCT magnets, the beam remains dynamically stable within the synchrotron. The study concludes that the reduction in dynamic aperture is predominantly determined by the physical aperture limits of the CCT bending magnets themselves, rather than the beam instability induced by high-order field nonlinearities. This innovative design demonstrates the strong potential of utilizing strongly curved CCT superconducting magnets to significantly reduce the footprint of heavy-ion therapy facilities, paving the way of the next generation of compact medical accelerators.