In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in (Formula presented.) and when this association forms a linear operator of norm-1 from (Formula presented.) to (Formula presented.). It is proved that, under certain geometric assumptions on (Formula presented.) these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if (Formula presented.) has property-U (SU) in (Formula presented.). We prove that when (Formula presented.) has the Radon–Nikod (Formula presented.) m Property for (Formula presented.), (Formula presented.) has property-U (property-SU) in (Formula presented.) if and only if Y is so in X. We show that if (Formula presented.) and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and (Formula presented.) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in (Formula presented.) which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of (Formula presented.) which have property-U (SU). © 2021 Informa UK Limited, trading as Taylor & Francis Group.