Some results of Kobayashi and M. G. Zajdenberg [Sib. Math. J. 24, 858-867 (1983); translation from Sib. Mat. Zh. 24, No. 6(142), 44-55 (1983; Zbl 0579.32039)] are completed to get the inverse statements. Let \(M\) be a locally complete hyperbolic and relatively compact subspace of a complex space \(X\). Then the following conditions are equivalent: i) \(M\) is hyperbolically embedded in \(X\), ii) \(M\) has an extension property for maps of punctured disk into \(M\subset X\), and \(\partial M\) contains no limit complex lines, iii) \(M\) contains no complex lines and \(\partial M\) contains no limit complex lines.