  Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every
infinite-dimensional $1$-injective Banach space contains a hyperplane that is
$(2+\varepsilon)$-injective for every $\varepsilon > 0$, yet is is \emph{not}
$2$-injective and remarked in a footnote that Pe{\l}czy\'nski had proved for
every $\lambda > 1$ the existence of a $(\lambda + \varepsilon)$-injective
space ($\varepsilon > 0$) that is not $\lambda$-injective. Unfortunately, no
trace of the proof of Pe{\l}czy\'nski's result has been preserved. In the
present paper, we establish the said theorem for $\lambda\in (1,2]$ by
constructing an appropriate renorming of $\ell_\infty$. This contrasts (at
least for real scalars) with the case $\lambda = 1$ for which Lindenstrauss
[Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.
