  Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it
is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining
norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$,
such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}}
\{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x\|=1\})\} >0.$ This shows that a
version of the Bishop-Phelps-Bollob\'as property in which the operator is not
changed is possible only if one of the involved Banach spaces is
one-dimensional.
