  We show that the probability, P_0(l), that the height of a fluctuating
(d+1)-dimensional interface in its steady state stays above its initial value
up to a distance l, along any linear cut in the d-dimensional space, decays as
P_0(l) \sim l^(-\theta). Here \theta is a `spatial' persistence exponent, and
takes different values, \theta_s or \theta_0, depending on how the point from
which l is measured is specified. While \theta_s is related to fractional
Brownian motion, and can be determined exactly, \theta_0 is non-trivial even
for Gaussian interfaces.
