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Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that everyinfinite-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 forevery $\lambda > 1$ the existence of a $(\lambda + \varepsilon)$-injectivespace ($\varepsilon > 0$) that is not $\lambda$-injective. Unfortunately, notrace of the proof of Pe{\l}czy\'nski's result has been preserved. In thepresent paper, we establish the said theorem for $\lambda\in (1,2]$ byconstructing an appropriate renorming of $\ell_\infty$. This contrasts (atleast for real scalars) with the case $\lambda = 1$ for which Lindenstrauss[Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.

A forgotten theorem of Pe{\l}czy\'nski: $(\lambda+)$-injective spaces need not be $\lambda$-injective -- the case $\lambda\in (1,2]$