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We study a variant of the Whitney extension problem for the space$C^k\Lambda^m_{\omega}(R^n)$ of functions whose partial derivatives of order$k$ satisfy the generalized Zygmund condition. We identify$C^k\Lambda^m_{\omega}(R^n)$ with a space of Lipschitz mappings from a metricspace $(R^{n+1}_+,\rho_\omega)$ supplied with a hyperbolic metric $\rho_\omega$into a metric space $({\cal P}_{k+m-1}\times R^{n+1}_+, d_\omega)$ ofpolynomial fields on $R^{n+1}_+$ equipped with a hyperbolic-type metric$d_\omega$. This identification allows us to reformulate the Whitney problemfor $C^k\Lambda^m_{\omega}(R^n)$ as a Lipschitz selection problem forset-valued mappings from $(R^{n+1}_+,\rho_\omega)$ into a certain family ofsubsets of ${\cal P}_{k+m-1}\times R^{n+1}_+$.

The Whitney extension problem for Zygmund spaces and Lipschitz selections in hyperbolic jet-spaces