In the course of reading a paper , I've encountered the following property of interest.

If $R$ is a ring, say it satisfies (*) if: For any smooth, irreducible $R$-algebra $B$ of finite type such that all the fibers of $Spec B$ over points of codimension one in $Spec R$ are irreducible, then $(B \otimes_R K)^* = B^* K^*$.

The author remarks that it is easy to verify property (*) for UFDs. I think I can sort of see the reasoning for PIDs: If $f \in (B \otimes_R K)^* $, we want to show that we may modify $f$ by some $c$ in $K^*$ such that for all primes $\mathfrak{p}$ of $(B \otimes_R K)$, $cf \notin \mathfrak{p}$. The image of the point $\mathfrak{p}$ under the map to $Spec R$ is some principal prime $(\alpha)$, so we modify by the appropriate power of $\alpha$.

What's a proof that UFDs satisfy (*)?