We present a fully implicit formulation of coupled fluid flow and geomechanics for fluid injection/withdrawal in fractured reservoirs in the context of storage. Utilizing a Galerkin finite‐element approach, both flow and poroelasticity equations are discretized on a shared three‐dimensional mesh. The fluid flow is assumed to be single‐phase. The hydraulic behaviour of fractures is represented through a double‐nodes flow element, which allows to efficiently model longitudinal and transversal fracture permeabilities. In addressing the mechanical sub‐problem, fractures are explicitly modelled using cohesive elements to account for contact, friction and opening phenomena. The non‐linear set of equations is solved implicitly through an iterative partitioned conjugate gradient procedure, extending its traditional application to continuous problems to those involving explicit discontinuities such as faults and fractures. The model's accuracy is verified against analytical solutions for different geomechanical problems, notably for the growth of a frictional slip rupture along a fault due to fluid injection. Such a particularly challenging benchmark for a critically stressed fault is here reproduced for the first time by a finite element–based scheme. The capabilities of the developed parallel solver are then illustrated through a scenario involving injection into a faulted aquifer. The original solver code, tutorials and data visualization routines are publicly accessible.