  In this paper we study the local times of vector-valued Gaussian fields that
are `diagonally operator-self-similar' and whose increments are stationary.
Denoting the local time of such a Gaussian field around the spatial origin and
over the temporal unit hypercube by $Z$, we show that there exists
$\lambda\in(0,1)$ such that under some quite weak conditions,
$\lim_{n\rightarrow +\infty}\frac{\sqrt[n]{\mathbb{E}(Z^n)}}{n^\lambda}$ and
$\lim_{x\rightarrow +\infty}\frac{-\log
\mathbb{P}(Z>x)}{x^{\frac{1}{\lambda}}}$ both exist and are strictly positive
(possibly $+\infty$).
  Moreover, we show that if the underlying Gaussian field is `strongly locally
nondeterministic', the above limits will be finite as well. These results are
then applied to establish similar statements for the intersection local times
of diagonally operator-self-similar Gaussian fields with stationary increments.
