The work in this paper delves into the challenge of whole elongated body's obstacle avoidance during path following for a class of bionic snake robots. Currently, most studies focus solely on preventing the robot's head from colliding with obstacles through designed controllers. However, due to the unique elongated structure and biomimetic locomotion modes of snake robots, it is unavoidable that the rest of the robot's body could still collide with obstacles. To resolve this problem, we propose a novel real-time optimization obstacle avoidance strategy for a class of terrestrial snake robots with multi-link elongated body using model predictive control (MPC). Moreover, by leveraging the elongated body characteristics of the robot, an improved path guidance strategy is also developed. The effectiveness of the proposed strategies is verified and validated through extensive simulations and experiments on a custom-built nine-link elongated snake robot. The results demonstrate that all links of the robot can well avoid obstacles while continuing to track the given path.