Context-free languages not closed under making them "extension-free"
Rather than the language $L\subseteq \Sigma^*$ consider the language $L' =L\$\$$, so concatenate every string by two copies of $\$$ where $\$$ is a new symbol not in $\Sigma$.

Let $x\in \Sigma^*$. 
String $x\$$ is not a proper prefix of $L'$ iff $x\$\$ \notin L'$ iff $x\notin L$.

That should start you going.