  Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study
Gorenstein derived functors for extriangulated categories. More precisely, we
first introduce the notion of the proper $\xi$-Gorenstein projective resolution
for any object in $\mathcal{C}$ and define the functors
$\xi\text{xt}_{\mathcal{GP}(\xi)}$ and $\xi\text{xt}_{\mathcal{GI}(\xi)}$.
Under some assumptions, we give some equivalent characterizations for any
object with finite $\xi$-Gorenstein projective dimension. Next we get some nice
results by using derived functors. As an application, our main results
generalize their work by Ren-Liu. Moreover, our proof is not far from the usual
module categories or triangulated categories.
