Let $M:=\{x\in\mathbb R^2:\left\|x\right\|\le1\}$, $\mathbb H^2:=\mathbb R\times[0,\infty)$ and $$\pi_p:\partial M\setminus\{x\in\partial M:x_2=p_2\}\to\partial\mathbb H^2\;,\;\;\;x\mapsto\left(x_1-x_2\frac{p_1-x_1}{p_2-x_2},0\right)$$ for $P\in\partial M$.
Can we form an atlas for $M$ (considered as a $C^\infty$-submanifold of $\mathbb R^2$ with boundary) using suitable extensions of the maps $\pi_p$?
By selecting two different $p\in\partial M\setminus\{x\in\partial M:x_2=p_2\}$, we can clearly cover $\partial M$ by the preimages of the $\pi_p$.