We address the key open problem of a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model. We construct a model on a lattice of SYK dots with nonrandom intersite hopping. The crucial feature of the resulting band dispersion is the presence of a Lifshitz point where two bands touch with a tunable power-law divergent density of states (DOS). For a certain regime of the power-law exponent, we obtain a class of interaction-dominated non-Fermi-liquid (NFL) states, which exhibits exciting features such as a zero-temperature scaling symmetry, an emergent (approximate) time reparameterization invariance, a power-law entropy-temperature relationship, and a fermion dimension that depends continuously on the DOS exponent. Notably, we further demonstrate that these NFL states are fast scramblers with a Lyapunov exponent lambda(L) proportional to T, although they do not saturate the upper bound of chaos, rendering them truly unique.