Accurately known stellar lithium abundances may be used to shed light on a variety of astrophysical phenomena such as big bang nucleosynthesis, radial migration, ages of stars and stellar clusters, and planet engulfment events. We present a grid of synthetic lithium spectra that are computed in non-local thermodynamic equilibrium (NLTE) across the STAGGER grid of three-dimensional (3D) hydrodynamic stellar atmosphere models. This grid covers three Li lines at 610.4, 670.8, and 812.6 nmfor stellar parameters representative of FGK-type dwarfs and giants, spanning T-eff = 4000-7000 K, log g = 1.5-5.0, [Fe/H] = -4.0-0.5, and A(Li) = -0.5-4.0. We find that our abundance corrections are up to 0.15 dex more negative than in previous work, due to a previously overlooked NLTE effect of blocking of UV lithium lines by background opacities, which has important implications for a wide range of science cases. We derive a new 3D NLTE solar abundance of A(Li) = 0.96 +/- 0.05, which is 0.09 dex lower than the commonly used value. We make our grids of synthetic spectra and abundance corrections publicly available through the BREIDABLIK package. This package includes methods for accurately interpolating our grid to arbitrary stellar parameters through methods based on Kriging (Gaussian process regression) for line profiles, and multilayer perceptrons (a class of fully connected feedforward neural networks) for NLTE corrections and 3D NLTE abundances from equivalent widths, achieving interpolation errors of the order of 0.01 dex.