  In this work we shall be concerned with some stability aspects of the
classical problem of extension of $C(K)$-valued operators. We introduce the
class $\mathscr{LP}$ of Banach spaces of Lindenstrauss-Pe{\l}czy\'{n}ski type
as those such that every operator from a subspace of $c_0$ into them can be
extended to $c_0$. We show that all $\mathscr{LP}$-spaces are of type $\mathcal
L_\infty$ but not the converse. Moreover, $\mathcal L_\infty$-spaces will be
characterized as those spaces $E$ such that $E$-valued operators from
$w^*(l_1,c_0)$-closed subspaces of $l_1$ extend to $l_1$. Complemented
subspaces of $C(K)$ and separably injective spaces are subclasses of
$\mathscr{LP}$-spaces and we show that the former does not contain the latter.
It is established that $\mathcal L_\infty$-spaces not containing $l_1$ are
quotients of $\mathscr{LP}$-spaces, while $\mathcal L_\infty$-spaces not
containing $c_0$, quotients of an $\mathscr{LP}$-space by a separably injective
space and twisted sums of $\mathscr{LP}$-spaces are $\mathscr{LP}$-spaces.
