Soft actuators that execute diverse motions have recently been proposed to improve the usability of soft robots. Nature-inspired actuators, in particular, are emerging as a means of accomplishing efficient motions based on the flexibility of natural creatures. In this research, we present an actuator capable of executing multi-degree-of-freedom motions that mimics the movement of an elephant’s trunk. Shape memory alloys (SMAs) that actively react to external stimuli were integrated into actuators constructed of soft polymers to imitate the flexible body and muscles of an elephant’s trunk. The amount of electrical current provided to each SMA was adjusted for each channel to achieve the curving motion of the elephant’s trunk, and the deformation characteristics were observed by varying the quantity of current supplied to each SMA. It was feasible to stably lift and lower a cup filled with water by using the operation of wrapping and lifting objects, as well as effectively performing the lifting task of surrounding household items of varying weights and forms. The designed actuator is a soft gripper that incorporates a flexible polymer and an SMA to imitate the flexible and efficient gripping action of an elephant trunk, and its fundamental technology is expected to be used as a safety-enhancing gripper that requires environmental adaptation.