Let $X,Y$ be Banach spaces, and let $X_0$ be a subspace of $X$ (by subspace I mean a closed linear set). Consider the set $Ext(X_0,Y)$ of all bounded linear operators $A_0:X_0\to Y$ which have an extension to a bounded linear operator $A:X\to Y$ (not necessarily with the same norm). Question: is $Ext(X_0,Y)$ closed in $B(X_0,Y)$ (with the usual operator norm)? Equivalently, is the set of all bounded linear operators $A_0:X_0\to Y$ which have no extension to a bounded linear operator $A:X\to Y$ open in $B(X_0,Y)$?

If $Ext(X_0,Y)$ is not closed (in general), for which $X,X_0;Y$ this set is closed in $B(X_0,Y)$?

I will be very grateful for any remarks and comments.