The following looks like a strengthening of the approximation property, but I don't know, maybe this is equivalent. I would be grateful if somebody could explain this.

Let $X$ be a Banach space with the approximation property, and $K$ a compact set in $X$.

Is it true that for each $\varepsilon>0$ there exists an (bounded) operator of finite rank $T:X\to X$ such that

1) $T$ approximates the identity operator on $K$: $$ \forall x\in K\qquad ||Tx-x||<\varepsilon, $$ and

2) the range of $T$ belongs to the linear span of $K$ $$ T(X)\subseteq\text{span}\ K $$ ?

EDIT. Yes, I see that this is not equivalent, because otherwise we obtain that the (usual) approximation property is inherited by closed subspaces. OK, I change a bit my question:

Is anything known about this variant of the approximation property?