Assume $V\vDash \mathsf{GCH}$, and let $G$ be the generic of Sacks forcing (also known as perfect tree forcing, see Jech ch. 15). Is it true $V[G]\vDash \mathsf{GCH}$?
It's pretty simple to show that $V[G]$ satisfies $2^{\aleph_\alpha}=\aleph_{\alpha+1}$ for all $\alpha > 0$, using the concept of 'nice names' [see the definition in Kunen VII 5.11]. In summary - there are at most $\aleph_2$ possible antichains, so at most $\aleph_2^\lambda$ different nice names for subsets of $\lambda$. For $\lambda\geq\aleph_1$ this is equal to $\lambda^+$, and so $(2^\lambda=\lambda^+)^{V[G]}$.
So the question really boils down to whether $V[G]\vDash\mathsf{CH}$?
Yes, Sacks forcing preserves $\mathsf{CH}$. This follows from the usual argument showing that the Sacks extension is minimal. One can read that proof as establishing that if $f$ is a new real (seen as a function from $\omega$ to $2$) then there is a continuous map in the ground model whose natural interpretation in the extension sends the Sacks generic to $f$.