  We give examples of infinitely extendable (not as cones) arithmetically
Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces
and which are not complete intersections. The proof uses the computation of the
dimension of the Hilbert scheme of codimension $2$ subschemes of projective
spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$
subschemes due to J. O. Kleppe and R.-M. Mir\'{o}-Roig.
