We propose and analyze a generalization of the Kitaev chain for fermions with long-range p-wave pairing, which decays with distance as a power law with exponent a. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range (a > 1) or purely algebraically (a < 1). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small a. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.