dg.differential geometry – Immersion of a part of the hyperbolic plane in $mathbb{R}^3$

I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert’s theorem ensures that there is no isometric immersion of the hyperbolic plane $mathbb{H}^2$ in $mathbb{R}^3$, but if we remove a point from the hyperbolic plane, can it be immersed isometrically in $mathbb{R}^3$? Who could it be?