The lattice Boltzmann method (LBM) has developed from a flow solver into a general discretization framework for problems in computational mechanics and applied mathematics. Its classical formulation on Cartesian lattices, however, fundamentally limits geometric fidelity, impeding the accurate treatment of curved domains and obstructing systematic connections with high-order geometry-based discretizations. We introduce an isogeometric lattice Boltzmann method (IGA-LBM) that incorporates spline-based geometry representations directly into the kinetic formulation. By mapping the discrete velocity lattice onto body-fitted B-spline and NURBS parameterizations, the approach reformulates the streaming and collision operators in curvilinear coordinates while preserving discrete conservation properties and the explicit structure characteristic of LBM. The resulting framework establishes a consistent link between kinetic schemes and isogeometric discretizations commonly used in computational mechanics. The higher-order continuity of spline spaces improves gradient resolution and reduces numerical dissipation near curved boundaries. The parametric setting further enables systematic h-, p- and k-refinement strategies, analogous to those in isogeometric analysis, and the diffeomorphic geometry mapping promotes intrinsic mass conservation and enhanced stability. Benchmark problems involving curved geometries demonstrate that IGA-LBM provides substantially improved predictions of wall stresses, pressure distributions, and aerodynamic forces compared with standard Cartesian LBM at comparable computational cost. Beyond fluid applications, the proposed formulation illustrates how kinetic solvers can be integrated with CAD-exact spline spaces, positioning IGA-LBM as a general-purpose high-order discretization paradigm for computational mechanics.