  By means of a family of counter-examples, it is shown that the Reilly upper
bound for the first eigenvalue of the Laplace operator for a compact
submanifold in Euclidean space does not work for $n$-dimensional compact
spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq
n+2$. We develop a new suitable technique, based on an integral formula on
compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family
of extrinsic upper bounds for the first eigenvalue of the Laplace operator for
a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each one of
these inequalities, becoming an equality can be characterized in geometric
terms. In particular, the eigenvalue achieves one of these upper bounds if and
only if the submanifold lies minimally in certain hypersphere of a spacelike
hyperplane.
