Is a local isometry of the hyperbolic plane the restriction of a global isometry?

Question

The origin question: Let $\Omega \subset \mathbb{H}^2$ be a domain of the hyperbolic plane $\mathbb{H}^2$. Let $u: \Omega \to \mathbb{H}^2$ be injective and an isometry from $\Omega$ to its image. Does there exist a Mobius transformation $\gamma\in \text{PSL}(2,\mathbb{R})$ such that $u=\gamma\mid_\Omega$?

The modified question: Let $\Omega \subset \mathbb{H}^2$ be a connected domain of the hyperbolic plane $\mathbb{H}^2$. Let $u: \Omega \to \mathbb{H}^2$ be an orientation-reserving $C^1$ isometry from $\Omega$ to its image. Does there exist a Mobius transformation $\gamma\in \text{PSL}(2,\mathbb{R})$ such that $u=\gamma\mid_\Omega$?

Thanks for all comments and answers. I have found the answer from "Dierkes, Ulrich; Hildebrandt, Stefan; Tromba, Anthony J. Global analysis of minimal surfaces", on page 273, Lemma 1, which reads as follows:

Lemma: Let $f: U \to \mathbb{H}^2$ be a $C^1$ isometry on an open connected subset $U$ of the hyperbolic plane. Then $$ f(w)=\frac{Aw+B}{Cw+D}, \, A,B,C,D \in \mathbb{R}, $$ and $AD-BC=1$.

Answer 1

Assuming that $\Omega$ is open, non-empty, and connected, then yes. In fact, an isometry determines, and is determined by, how it transforms any one point and any one "frame" (orthonormal basis) at that point.

Answer 2

The hyperbolic plane has this property, as does the Euclidean plane.

If $K$ is any subset of $ \mathbb{H}^2$, and $u : K \to\mathbb{H}^2$ is an isometry, then there is an extension of $u$ which is an isometry of $\mathbb{H}^2$ onto itself. I use "isometry" in the sense: $d(u(x),u(y)) = d(x,y)$ for all $x,y \in E$, where $d$ is the hyperbolic distance in $\mathbb{H}^2$.

So the OP reduces to: is an isometry of $\mathbb{H}^2$ onto itself necessarily a Möbius transformation (or the conjugate of a Möbius transformation)?