Lifting invertible functions on a divisor to ambient affine variety Let $V$ be an affine variety over an algebraically closed field $k$ and 
$D \subset V$ a Cartier divisor which is normal and has an isolated singularity at $p \in D$.
Let $\mathcal{O}_V^*, \mathcal{O}_D^*$ be the sheaves of invertible functions on $V$ and $D$. 
Then I think that we have an exact sequence $0 \rightarrow K \rightarrow \mathcal{O}_V^* \rightarrow \mathcal{O}_D^* \rightarrow 0$. 
Question (Edited) Is there an affine open neighbourhood of $p \in V' \subset V$ such that $H^0(V', \mathcal{O}_{V'}^*) \rightarrow H^0(D', \mathcal{O}_{D'}^*)$ is surjective where $D':= D \cap V'$? That is, can we lift a surjection of stalks to that on some open neighbourhood?
I think the Question is reduced to the following. 
Question' Is the cokernel of $H^0(V, \mathcal{O}_{V}^*) \rightarrow H^0(D, \mathcal{O}_{D}^*)$ finitely generated as an abelian group? 
 A: Suppose that $\bar{f} \in H^0(D, O_D^*)$ and consider a corresponding $f \in H^0(V, O_V)$ (which may or may not be invertible).  
Then for every point $x \in D \subseteq V$, we let $\bar{f}'$ denote the element in the stalk $O_{D,x}$ and $f'$ the element in the stalk $O_{V,x}$.   Since $\bar{f}'$ is not in the maximal ideal of $O_{D,x}$, neither is $f'$ in the maximal ideal of $O_{V,x}$.  Thus $f'$ is invertible in a neighborhood of $x \in V$.  Since this holds for all points $x \in D$, the vanishing locus $V(f')$ of $f'$ is away from $D$.  It follows that there exists a neighborhood of $D$ where $f'$ is a unit.
Now, I just learned from 
THIS QUESTION
(that at least in the geometric setting you are interested in) the set of units of $H^0(D, O_D)$ is finitely generated modulo constants.  Thus, choose generators ${\bar f_1}, \dots, \bar{f_n}$ of $H^0(D, O_D^*)$ modulo constants.  Lifting these to $f_i \in H^0(V, O_V)$, we can find an open set $U \subseteq V$ containing $D$ such that the $f_i$ are invertible in $H^0(U, O_U)$.  It follows that $H^0(U, O_U^*) \to H^0(D, O_D^*)$ is surjective.
Thus it seems we can get a slightly stronger statement than what you asked for.
Statement: $\text{ }$ There exists an open neighborhood $U \subseteq X$ containing $D$ such that $H^0(U, O_U^{*}) \to H^0(V, O_V^{*})$ is surjective.
EDIT: Perhaps in view of the newly revised question which appeared while I was typing this (the finite generation part), this is more information than required.  But perhaps it will be useful to someone.
A: Edit:  As Jason points out, the following answers the original question, but not the revised question.
Let $E$ be an elliptic curve in $\mathbb{P}^2_{\mathbb{C}}$ and $P\in E$ a point of order $2$.  The tangent line $L$ to $E$ at $P$ meets $E$ at $P$ and the identity $O$.  Now $E\setminus L$ is a divisor in $\mathbb{A}^2_{\mathbb{C}} = \mathbb{P}^2_{\mathbb{C}}\setminus L$ that carries a non-constant invertible function.
