  We introduce a notion of intrinsically Lipschitz graphs in the context of
metric spaces. This is a broad generalization of what in Carnot groups has been
considered by Franchi, Serapioni, and Serra Cassano, and later by many others.
We proceed by focusing our attention on the graphs as subsets of a metric space
given by the image of a section of a quotient map and we require an
intrinsically Lipschitz condition. We shall not have any function on a
topological product, not we shall consider a metric on the base of the quotient
map. Our results are: an Ascoli-Arzel\`a compactness theorem, an Ahlfors
regularity theorem, and some extension theorems for partially defined
intrinsically Lipschitz sections. Known results by Franchi, Serapioni, and
Serra Cassano, and by Vittone will be our corollaries.
