We investigate the tip region of a hydraulic fracture driven by a power law fluid propagating at a constant velocity in an impermeable linear elastic isotropic medium. We account for the presence of a fluid lag of a priori unknown length at the tip of the fracture. The fluid pressurized fracture propagates perpendicular to the far-field compressive minimum stress in pure opening mode. The solution of this semi-infinite hydraulic fracture problem combines elastic deformation, lubrication flow in the filled region of the fracture and the quasi-static equilibrium fracture propagation condition. It exhibits a multiscale structure related to the strong fluid solid coupling at play near the fracture tip. For a Newtonian fluid, the solution derived by Garagash & Detournay (2000) transition from the classical linear elastic fracture mechanics asymptote near the fracture tip to the viscous asymptote (Desroches et al., 1994) away from the tip. The fluid lag was shown to vanish for large values of a dimensionless fracture toughness encapsulating all the problem parameters (viscosity, fracture velocity, plane- strain elastic modulus).