  It is hereby established that the set of Lipschitz functions
$f:\mathcal{U}\rightarrow \mathbb{R}$ ($\mathcal{U}$ nonempty open subset of
$\ell_{d}^{1}$) with maximal Clarke subdifferential contains a linear subspace
of uncountable dimension (in particular, an isometric copy of
$\ell^{\infty}(\mathbb{N})$). This result goes in the line of a previous result
of Borwein-Wang. However, while the latter was based on Baire category theorem,
our current approach is constructive and is not linked to the uniform
convergence. In particular we establish lineability (and spaceability for the
Lipschitz norm) of the above set inside the set of all Lipschitz continuous
functions.
