Let $X=\{(X_1(t),\dots,X_d(t)): t\in \mathbb{R}^n\}$ be a Gaussian random field in $\mathbb{R}^d$ such that $X_1,\dots,X_d$ are independent, centered Gaussian random fields with continuous sample paths. Let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map and let $A\subset \mathbb{R}^n$ be an analytic set. The main goal of the paper is to determine the almost sure value of the packing dimension of the image and graph of $X+f$ restricted to $A$ under a very mild assumption. This generalizes a result of Du, Miao, Wu and Xiao, who calculated the packing dimension of $X(A)$ if $X_1,\dots,X_d$ are independent copies of the same Gaussian random field $X_0$. Provided that $X$ is a fractional Brownian motion, our result is new even if $n=d=1$ and $f$ is continuous, and even if $f\equiv 0$ in the case of graphs. For a fractional Brownian motion $X$ we also obtain the sharp lower bound for the packing dimension of the graph of $X$ over $A$ in terms of the Hurst index of $X$ and the packing dimension of $A$. The analogous result for images was obtained by Talagrand and Xiao.