The present invention provides a flexible suction body and method of its use for temporarily immobilizing a local area of tissue during surgery or a diagnostic procedure. The invention device and method are particularly well suited for immobilizing a local area of heart tissue to thereby permit surgery, for example, minimally invasive or micro-surgery, on a coronary vessel without significant deterioration of the pumping function of the beating heart. The invention suction device, which is coupled to a vacuum source, has a ring- or horseshoe-shaped suction channel with a flexible, spreadable outer rim that flattens and spreads outwardly against the surface of the treatment site as air is withdrawn from the suction channel. This design distributes the pressure of the suction body against the treatment site while maximizing the suction field to minimize bruising and suction hemorrhage at the treatment site caused by the suction body. In one embodiment, the suction body is sized for attachment to the tip of a hollow endoscopic catheter, with the hole in the center of the suction body being mounted around the periphery of the hollow catheter tip and allowing for passage from the catheter into the treatment site of a needle, or trocar for injecting materials such as drugs or genetic constructs into the treatment site or for piercing a treatment site, such as a solid tumor, to remove fluid or tissue therefrom for diagnosis. The suction body is preferably manufactured of a soft, compliant material, such as an elastomeric polymer, to conform to the surface of the heart and may be contoured in overall shape to further enhance its conformation to a contoured surface, such as the surface of the heart.