We introduce a class of variational states to study ground-state properties and real-time dynamics in (2 + 1)-dimensional compact QED. These are based on complex Gaussian states which are made periodic to account for the compact nature of the U (1) gauge field. Since the evaluation of expectation values involves infinite sums, we present an approximation scheme for the whole variational manifold. We calculate the ground-state energy density for lattice sizes up to 20 × 20 and extrapolate to the thermodynamic limit for the whole coupling region. Additionally, we study the string tension both by fitting the potential between two static charges and by fitting the exponential decay of spatial Wilson loops. As the ansatz does not require a truncation in the local Hilbert spaces, we analyze truncation effects which are present in other approaches. The variational states are benchmarked against exact solutions known for the one plaquette case and exact diagonalization results for a Z3 lattice gauge theory. Using the time-dependent variational principle, we study real-time dynamics after various global quenches, e.g., the time evolution of a strongly confined electric field between two charges after a quench to the weak-coupling regime. Up to the points where finite-size effects start to play a role, we observe equilibrating behavior.