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.