The structural and electronic properties of beryllium substitutional acceptors and interstitial donors in GaN are investigated using first-principles calculations based on pseudopotentials and density-functional theory. In p-type GaN, Be interstitials, which act as donors, have formation energies comparable to that of substitutional Be on the Ga site, which is an acceptor. In thermodynamic equilibrium, incorporation of Be interstitials will therefore result in severe compensation. To investigate the kinetics of Be interstitial incorporation and outdiffusion we have explored the total-energy surface. The diffusivity of Be interstitials is highly anisotropic, with a migration barrier in planes perpendicular to the c axis of 1.2 eV, while the barrier for motion along the c axis is 2.9 eV. We have also studied complex formation between interstitial donors and substitutional acceptors, and between hydrogen and substitutional beryllium. The results for wurtzite GaN are compared with those for the zinc-blende phase. Consequences for p-type doping using Be acceptors are discussed.