  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 metric
space $(R^{n+1}_+,\rho_\omega)$ supplied with a hyperbolic metric $\rho_\omega$
into a metric space $({\cal P}_{k+m-1}\times R^{n+1}_+, d_\omega)$ of
polynomial fields on $R^{n+1}_+$ equipped with a hyperbolic-type metric
$d_\omega$. This identification allows us to reformulate the Whitney problem
for $C^k\Lambda^m_{\omega}(R^n)$ as a Lipschitz selection problem for
set-valued mappings from $(R^{n+1}_+,\rho_\omega)$ into a certain family of
subsets of ${\cal P}_{k+m-1}\times R^{n+1}_+$.
