Let $l^{\infty}$ be the Banach space of all bounded real sequences with the $sup$norm and $c$ the closed subspace of convergent sequences. Is there a continuous linear map $T: l^{\infty} \rightarrow c$ such that $T$ is the identity on $c$?

Such $T$ does not exist because $c_0$ is not complemented in $l_\infty$ but it is complemented in $c$. See for example "Topics in Banach Space Theory" by Kalton and Albiac. 

