I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ2 into ℝ3. However, a perturbation of the pseudosphere, Dini's surface (or rather family of surfaces), which are isometrically embedded one-sided tubular neighborhoods of a geodesic in the hyperbolic plane (see https://mathoverflow.net/a/149884/1345), seems to do the trick since they contain arbitrarily large disks in the hyperbolic plane by varying the parameter so that the radius of the tubular neighborhood grows. See Dini's Surface at the Geometry Center.
