Truss-based lattice materials are cellular materials with an outstanding potential for multi-functional use. This is owing to properties of high compressive strength to density ratios combined with a periodic and open structure. However, such structures at low relative densities are particularly vulnerable to elastic buckling failure. Fibre-reinforcement that increases the buckling strength of lattice materials is proposed and the behaviour of unit cells that are tessellated within the lattice is investigated. A two-dimensional square orientated unit cell and a three-dimensional tetrahedron-shaped unit cell are both modelled discretely using energy principles with the nonlinear interactive buckling behaviour being analysed. The analytical approach, based on a perturbation method, exhibits excellent agreement for the mechanical response when compared to results from numerical continuation for moderately large displacements. A fundamental understanding of the mechanical behaviour of a unit cell can be upscaled in future work. It is postulated that this will enable the determination of the constitutive behaviour of such lattice materials.