Proof in short: If we can construct a divisible cyclic submodule, it must be isomorphic to $R$, and therefore $R$ is a field. We will do this by showing that any element of a minimal set of generators of a divisible $R$-module generates a cyclic divisible submodule by showing that otherwise we obtain a smaller set of generators.
Proof: Suppose $x_1,\ldots, x_n$ is a minimal set of generators for a divisible $R$-module $M$.
First suppose $n=1$, so $M=Rx$ for some $x\in M$. Then for any $r\in R$, $x=ry$ for some $y\in M$; since $M$ is generated by $x$, we have $y=r'x$ for some $r'\in R$ and so $x=rr'x$. In particular, the map $r\mapsto rx$ is a $R$-module isomorphism, since $rx=0$ implies $0=r'rx=x$. In particular, we see that $r'r=1$ in $R$, so $r'=r^{-1}$, and thus $R$ is a field.
Now suppose $n>1$. Let $a\in R^\times$ and let $ay=x_1$. Then $y=r_1x_1+\ldots +r_nx_n$ for some $r_1,\ldots, r_n$; if such an expression exists where $r_2,\ldots, r_n$ are all zero for every $r$, then the submodule generated by $x_1$ is divisible and we reduce to the previous case.
Therefore, let us assume that for some $r\in R$, in any such expression for $y$ we have some $r_m\neq 0$ for $m>1$. We will show that we can pick a smaller generating set, contradicting our assumption on the generators.
We have $x_1=ar_1x_1+ar_2x_2+\ldots+ar_nx_n$. Note we cannot have $r_1=a^{-1}$, since then we could write $x_1=ar_1x_1$, contradicting our assumption that some $r_m\neq 0$ for $m>1$. Since $ar_1\neq 1$, $\lambda=1-ar_1\neq 0$ in $R$, so $x_1=\lambda (s_1x_1+\ldots s_nx_n)$ for some $s_1,\ldots, s_n\in R$. But then
$$x_1=s_1\lambda x_1+s_2\lambda x_2+\ldots +s_n \lambda x_n=(s_2\lambda+as_1r_2)x_2+\ldots+(s_n\lambda+as_1r_n)x_n,$$
hence $M$ is generated by $x_2,\ldots, x_n$ contradicting our assumption.