A Banach space $X$ is said to have the {\it separable
lifting property} if for every subspace $Y$ of $X^{**}$
containing $X$ and such that $Y/X$ is separable there exists a
bounded linear lifting from $Y/X$ to $Y$. We show that if a
sequence of Banach spaces $E_1, E_2, \ldots$ has the joint
uniform approximation property and $E_n$ is $c$-complemented
in $E_n^{**}$ for every $n$ (with $c$ fixed), then $\eco$ has
the separable lifting property. In particular, if $E_n$ is a
${\mathcal{L}}_{p_n, \lambda}$-space for every $n$ ($1 <
p_n < \infty$, $\lambda$ independent of $n$), an $L_\infty$
or an $L_1$ space, then $\eco$ has the separable lifting
property. We also show that there exists a Banach space $X$
which is not extendably locally reflexive; moreover, for every
$n$ there exists an $n$-dimensional subspace $E \hra X^{**}$
such that if $u : X^{**} \raw X^{**}$ is an operator ($=$
bounded linear operator) such that $u (E) \subset X$, then
$||(u|_E)^{-1}|| \cdot ||u|| \geq c \sqrt{n}$, where $c$ is a
numerical constant.