Can the Poincare half plane be isometrically embedded into $\mathbb{R}^3$ (with Euclidean metric)?

Question

Can the Poincaré half plane be isometrically embedded into $\mathbb{R}^3$ (with Euclidean metric)? I am self-studying Riemannian geometry and have found it harder to imagine a surface with negative constant curvature, as opposed to a surface with positive constant curvature ($S^2$). That's why I have this question. It seems the Poincaré plane can be embedded into $\mathbb{R}^3$ with a Minkowski metric. I am quite curious if the Euclidean metric is possible.