As shown by David, the answer is **no** for line bundles. 

More generally, the answer in **no** even for higher dimensional vector bundles. In fact, in his paper "[Vector bundle over affine surfaces birationally equivalent to ruled surfaces][1]" Murthy proves the following results:

**(1)** Let $V$ be an irreducible affine non-singular surface defined over an algebraically closed field $k$ and such that $V$ is birational to $C \times \mathbb{P}^1$, vhere $C$ is a curve. Then any vector bundle over $V$ is a direct sum of a trivial bundle and a line bundle. In general, there are infinitely many equivalence classes of line bundles (as shown by David  for $V= \mathbb{P}^1 \times \mathbb{P}^1 \setminus \Delta$).

**(2)** There exist an affine, nonsingular rational variety of dimension $3$ over which there are infinitely many non-isomorphic *indecomposable* vector bundles of rank $>1$. 


  [1]: http://www.jstor.org/pss/1970667