We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y ) has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space Y . We show that in this case, there exists a universal function η X (ε) such that for every Y , the pair (X, Y ) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y ) has the Bishop-Phelps-Bollobás property for every Banach space X. In this case, we show that there is a universal function η Y (ε) such that for every X, the pair (X, Y ) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollobás property for c 0 -, 1 -and ∞ -sums of Banach spaces.goal here is to introduce and study analogues of properties A and B in the context of vector-valued versions of the Bishop-Phelps-Bollobás theorem.Let us now restart the Introduction, this time giving the necessary background material to help make the paper entirely accessible. The Bishop-Phelps-Bollobás property was introduced in 2008 [2] as an extension of the Bishop-Phelps-Bollobás theorem to the vector-valued case. It can be regarded as a "quantitative version" of the study of norm-attaining operators initiated by J. Lindenstrauss in 1963. We begin with some notation to present its definition. Let X and Y be Banach spaces over the field K = R or C. We will use the common notation S X , B X , X * for the unit sphere, the closed unit ball and the dual space of X respectively, L(X, Y ) for the Banach space of all bounded linear operators from X into Y, and NA(X, Y ) for the subset of all norm-attaining operators. (We say that an operator T ∈ L(X, Y ) attains its norm if T = T x for some x ∈ S X .) We will abbreviate L(X, X), resp. NA(X, X), by L(X), resp. NA(X). Definition 1.1 ([2, Definition 1.1]). A pair of Banach spaces (X, Y ) is said to have the Bishop-Phelps-Bollobás property (BPBp for short) if for every ε ∈ (0, 1) there is η(ε) > 0 such that for every T 0 ∈ L(X, Y ) with T 0 = 1 and every x 0 ∈ S X satisfying T 0 (x 0 ) > 1 − η(ε), there exist S ∈ L(X, Y ) and x ∈ S X such that
