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

Extending infinitely many times arithmetically Cohen-Macaulay and Gorenstein subvarieties of projective spaces