  In this article, we consider a stochastic partial differential equation
(SPDE) driven by a L\'evy white noise, with Lipschitz multiplicative term
$\sigma$. We prove that under some conditions, this equation has a unique
random field solution. These conditions are verified by the stochastic heat and
wave equations. We introduce the basic elements of Malliavin calculus with
respect to the compensated Poisson random measure associated with the L\'evy
white noise. If $\sigma$ is affine, we prove that the solution is Malliavin
differentiable and its Malliavin derivative satisfies a stochastic integral
equation.
