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I didn't see the exact answer to your question in the Bobiskenko paper, since section 2.4 only seems to address immersions of subsets of H^2 into E^3. However, a perturbation of the pseudosphere, Dini's surface, which is an embedded horocycle seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.