The following is from Rasmussen and Williams (2006):
A simple example of a Gaussian process can be obtained from our Bayesian linear regression model $f(x)=\phi(x)^Tw$ with prior $w\sim\mathcal N(0,\Sigma_p)$. We have for the mean and covariance $$ \begin{aligned} \mathbb E[f(x)]&=\phi(x)^T\mathbb E[w]=0\\ \mathbb E[f(x)f(x')]&=\phi(x)^T\mathbb E[ww^T]\phi(x')=\phi(x)^T\Sigma_p\phi(x') \end{aligned} $$ Thus, $f(x)$ and $f(x')$ are jointly Gaussian with zero mean and covariance given by $\phi(x)^T\Sigma_p\phi(x')$. Indeed, the function values $f(x_1),f(x_2),\ldots,f(x_n)$ corresponding to any number of input points $n$ are jointly Gaussian, although if $N<n$ then this Gaussian is singular (as the joint covariance matrix will be of rank $N$).
The joint distribution of $f(x_1),f(x_2),\ldots,f(x_n)$ is degenerate if $n$ is larger than the dimension of $w$. Can we still say then that $f(x_1),f(x_2),\ldots,f(x_n)$ are jointly Gaussian? Can $f(x)$ still be called a Gaussian process?