We present a novel poro-damage-viscoelastic model for predicting the failure response of fluid-saturated porous geomaterials. The Generalized Maxwell model is introduced for the representation of the viscoelastic behavior of the solid skeleton, which is achieved by a standard Prony-series type expansion. Damage regularization is obtained by an non-local integral-type formulation and damage behavior is described by Mazars model with the modified von Mises-type equivalent strain measure. The poromechanics parameters (Biot's coefficient, Biot's modulus) are functions of damage, and the fluid flow obeys Darcy's seepage law in the entire domain, while the permeability is assumed to be anisotropic and strain dependent. The coupled system is discretized in time using a backward Euler scheme. The non-linear system is linearized using a Newton Raphson scheme and solved monolithically every time step. A consistent Jacobian matrix and residual vector are derived analytically. Several numerical examples are studied in order to investigate the performance of the proposed approach, including (i) a column undergoing hysteresis from cyclic loading, stress relaxation, creep and variable strain rate loading tests and (ii) fluid-driven fracturing in a 2D poro-viscoelastic domain. The numerical time-dependent results exhibit mesh insensitivity for all field variables, and confirm the feasibility and applicability of the proposed non-local damage model for simulating hydraulic fracture.