Proper holomorphic map from the unit disc to the polydisc of dimension greater than one .

Question

Let $F:\mathbb{D}\rightarrow\mathbb{D}^n$ be a proper holomorphic map and $n\geq2.$ I have the following questions:

1. Does F extends in a nbhd of $\overline{\mathbb{D}}$ holomorphically?

2. If not what about the case when $F$ is injective?

Answer 1

No. Take for example $$F(z)=(z,g(z)),$$ where $g$ is any holomorphic function $\mathbb{D}\to\mathbb{D}$ that doesn't extend. This $F$ is both proper and injective.