In this paper, we introduce a new class of mapping: p-affinely orthogonal mapping, obtain some relations with isometric mapping, give some sufficient conditions under which an isometric mapping from unit sphere of L p ( μ ) into (or onto) that of another Banach space can be extended to be a linear isometry on the whole space, as well as show that any into (or onto) isometric mapping between the spheres of L p ( μ ) and L p ( ν , H ) ( 1 < p ≠ 2 , H is a Hilbert space) can be extended to be a real linear isometry on the whole space. Therefore, we generalize the corresponding results of [G.-G. Ding, The isometric extension problem in the unit spheres of l p ( Γ ) ( p > 1 ) type spaces, Sci. China Ser. A 46 (3) (2003) 333–338] and [J. Wang, On extension of isometries between unit spheres of AL p -spaces ( 0 < p < ∞ ) , Proc. Amer. Math. Soc. 132 (10) (2004) 2899–2909].