We present non-Born-Oppenheimer calculations of the electrical static polarizability of the LiH molecule. This is the first calculation of the non-Born-Oppenheimer polarizability for a system with more than two electrons. The polarizability is calculated by numerical differentiation of the energy obtained at different electric field strengths. The values for the energy are obtained by variational optimization with analytical gradients of the wave function expanded in a basis of explicitly correlated Gaussian functions. We also present a derivation of the integrals and energy gradients needed to perform these calculations. The result for the polarizability is 29.57 a.u.