In this article, we construct a Malliavin derivative for functionals of a square-integrable Lévy process. The Malliavin derivative is defined via chaos expansions involving mixed stochastic integrals with respect to the Brownian motion and the Poisson random measure. Some properties of this derivative are studied and a Clark-Ocone formula is derived. The construction and the results extend those for Brownian motion and pure-jump Lévy processes. Moreover, the explicit martingale representation of the maximum of a Lévy process is computed.