  We study a Bishop-Phelps-Bollob\'as version of Lindenstrauss properties A and
B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the
Bishop-Phelps-Bollob\'as property (BPBp) for every Banach space $Y$. We show
that in this case, there exists a universal function $\eta_X(\varepsilon)$ 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-Bollob\'as property for every Banach space
$X$. In this case, we show that there is a universal function
$\eta_Y(\varepsilon)$ 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-Bollob\'as property for $c_0$-, $\ell_1$- and
$\ell_\infty$-sums of Banach spaces.
