Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$ (with trivial generic stabilizer if necessary).

This question is about "hyperbolicity" and it is motivated by Lang's conjecture on integral points of varieties.

Assume that any atlas $U\to X$ (with $U$ a smooth quasi-projective variety and $U\to X$ etale surjective) is hyperbolic, i.e., any holomorphic map $\mathbb C\to U^{an}$ is constant.

Question. Is $X$ ``hyperbolic", i.e., is any holomorphic map $\mathbb C\to X^{an}$ constant?

The answer is yes if $X$ has a finite etale atlas $U\to X$, e.g., $X$ is the moduli stack of curves of genus at least two, cubic threefolds, polarized K3 surfaces, or polarized abelian varieties. The reason being that a holomorphic map $\mathbb C\to X^{an}$ lifts to a holomorphic map $\mathbb C\to U^{an}$ in this case.

But I'm sure there are DM-stacks with no finite etale atlas. There might have even been an MO question about this.

One result towards the existence of a finite atlas is Theorem 1.6.6 in Laumon--Moret-Bailly which states that there is a variety $Z$ and a finite surjective generically etale morphism $Z\to X$.

For orbifold curves we know that there is a finite etale atlas, so if $X$ is one-dimensional with trivial generic stabilizer the answer to the above question is positive.

Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.

Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?

What if $X$ is an algebraic space (i.e., trivial stabilizers)?

Edit: I changed the old question to a different question which should be more clear. An answer to the new question would help a lot in answering the old question.