We study the coherence of p-polarized light scattered from a one-dimensional weakly rough random metal surface in contact with vacuum. The mutual coherence function of the single nonzero component of the scattered magnetic field is calculated in planes parallel to, and at increasing distances from, the mean scattering surface in the vacuum region. It is found to be the sum of a contribution that is independent of the distance from the mean surface and a contribution that is a function of this distance and decays to zero over a distance of the order of the wavelength of the incident light. It is also shown that the spatial coherence of the electromagnetic field in the far field in a plane at a fixed distance from the mean surface, as a function of the relative distance along it, mimics the surface height autocorrelation function at short relative distances and oscillates with two periods, T(1) = lambda and T(2) = lambda/sin theta(0), where theta(0) is the angle of incidence. The former is due to the excitation of lateral waves, while the latter is due to the coherent interference of the multiple scattering processes that lead to the enhanced backscattering effect. In the near field the spatial coherence of the electromagnetic field measured at a fixed distance from the mean surface displays oscillations that are due to the excitation of surface plasmon polaritons. The period of these oscillations equals the wavelength of the surface plasmon polaritons, while the exponential decay of their amplitude is determined by the energy mean free path of the surface plasmon polaritons.