  We derive exact asymptotics of $$\mathbb{P}\left(\sup_{\mathbf{t}\in
{\mathcal{A}}}X(\mathbf{t})>u\right),~ \text{as}~ u\to\infty,$$ for a centered
Gaussian field $X(\mathbf{t}),~ \mathbf{t}\in \mathcal{A}\subset\mathbb{R}^n$,
$n>1$ with continuous sample paths a.s., for which $\arg \max_{\mathbf{t}\in
{\mathcal{A}}} Var(X(\mathbf{t}))$ is a Jordan set with finite and positive
Lebesque measure of dimension $k\leq n$ and its dependence structure is not
necessarily locally stationary. Our findings are applied to deriving the
asymptotics of tail probabilities related to performance tables and chi
processes where the covariance structure is not locally stationary.
