Marine gas hydrate formations are characterized by considerable water depth, shallow subsea burial, loose strata, and low formation temperatures. Drilling in such formations is highly susceptible to hydrate dissociation, leading to gas invasion, wellbore instability, reservoir subsidence, and sand production, posing significant safety challenges. While previous studies have extensively explored multiphase flow dynamics between the formation and the wellbore during conventional oil and gas drilling, a clear understanding of wellbore stability under the unique conditions of gas hydrate formation drilling remains elusive. Considering the effect of gas hydrate decomposition on formation and reservoir frame deformation, a multi-field coupled mathematical model of seepage, heat transfer, phase transformation, and deformation of near-wellbore gas hydrate formation during drilling is established in this paper. Based on the well logging data of gas hydrate formation at SH2 station in the Shenhu Sea area, the finite element method is used to simulate the drilling conditions of 0.1 MPa differential pressure underbalance drilling with a borehole opening for 36 h. The study results demonstrate a significant tendency for wellbore instability during the drilling process in natural gas hydrate formations, largely due to the decomposition of hydrates. Failure along the minimum principal stress direction in the wellbore wall begins to manifest at around 24.55 h. This is accompanied by an increased displacement velocity of the wellbore wall towards the well axis in the maximum principal stress direction. By 28.07 h, plastic failure is observed around the entire circumference of the well, leading to wellbore collapse at 34.57 h. Throughout this process, the hydrate decomposition extends approximately 0.55 m, predominantly driven by temperature propagation. When hydrate decomposition is taken into account, the maximum equivalent plastic strain in the wellbore wall is found to increase by a factor of 2.1 compared to scenarios where it is not considered. These findings provide crucial insights for enhancing the safety of drilling operations in hydrate-bearing formations.