When treating prostate cancer, the use of puncture robots is an effective method to perform radioactive seed implantation surgery. However, when the puncture needle enters the lesion, the soft tissue is easily deformed owing to the complex force between the puncture needle and soft tissue, which leads to a puncture deviation between the needle tip and target point. To solve this problem effectively, the prostate soft tissue puncture process is studied based on the analysis of the puncture needle–soft tissue interaction. First, the puncture force is classified into contact, friction, and cutting forces by a quantitative decomposition method, and the corresponding force model is established. Based on the theoretical analysis of the model, it is deduced that these factors can affect the deformation created by puncturing the soft tissue. Subsequently, a puncture platform is built and many biomimetic soft tissue models are established. Multiple puncture experiments on the influencing factors are conducted using the method of controlling a single variable. Using the spatial puncture deviation as the test metrics, the significance of the influencing factors of the puncture deformation is verified. Finally, it can be concluded from the experimental analysis that the main factor that affects the puncture deviation is the puncture speed, whereas the puncture depth has no significant influence. The puncture speed was optimized and verified by experiments, and the results showed that a stable puncture accuracy under different puncture depths can be obtained by selecting an optimized puncture speed (12.6 mm/s). This work provides a design reference to study the positioning accuracy of minimally invasive puncture surgery.