We construct and characterize tight-binding Hamiltonians which contain a completely flat topological band made of continuum lowest Landau-level wave functions sampled on a lattice. We find an infinite family of such Hamiltonians, with simple analytic descriptions. These provide a valuable tool for constructing exactly solvable models. We also implement a numerical algorithm for finding the most local Hamiltonian with a flat Landau level. We find intriguing structures in the spatial dependence of the matrix elements for this optimized model. The models we construct serve as foundations for numerical and experimental studies of topological systems, both noninteracting and interacting. © 2020 American Physical Society.