Split embedding problems over the open arithmetic disc

Let $\mathbb{Z}\{t\}$ be the ring of arithmetic power series that converge on the complex open unit disc. A classical result of Harbater asserts that every finite group occurs as a Galois group over the quotient field of $\mathbb{Z}\{t\}$. We strengthen this by showing that every finite split embedding problem over $\mathbb{Q}$ acquires a solution over this field. More generally, we solve all $t$-unramified finite split embedding problems over the quotient field of $\mathcal {O}_K\{t\}$, where $\mathcal {O}_K$ is the ring of integers of an arbitrary number field $K$.