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 &#8461;<sup>2</sup> into &#8477;<sup>3</sup>. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See [Dini's Surface][1] at the Geometry Center. 

![alt text][2]


  [1]: http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/dini/
  [2]: http://paulbourke.net/geometry/dini/dini3.gif