Hydraulic fracturing is an industrial process often applied to enhance oil and gas recovery. Under this process, fractures are generated by the injection of highly pressurized fluids, which often exhibit shear-thinning rheology and yield stress. The global fracture propagation is influenced by various processes occurring near the fracture tip. To gain an insight into fracture propagation, the problem of a semi-infinite hydraulic fracture propagating in a permeable linear elastic rock is solved. To investigate the effect of fluid yield stress, we focus on a fracture driven by Herschel–Bulkley fluid. The mathematical model consists of the elasticity equation, the lubrication equation, and the propagation criterion for the semi-infinite plane strain fracture to obtain the fracture opening. The non-linear system of governing equations is represented in the non-singular form and solved numerically using Newton’s method. The solution is influenced by the competing processes related to rock toughness, fluid properties, and leak-off. The effects of these phenomena prevail at different length scales, and the corresponding limits can be described via analytical solutions. For a Herschel-Bulkley fluid, an additional limiting solution related to the fluid yield stress is obtained, and the regions of the dominance of limiting solutions affected by the yield stress are investigated. Finally, a faster approximate solution for the problem is proposed and its accuracy against a numerical solution is evaluated. The obtained result can be applied in hydraulic fracturing simulators to account for the effect of Herschel–Bulkley fluid rheology on the near-tip region.