Every Dunford-Pettis operator is compact

Question

I was trying this problem.

Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every Dunford-Pettis operator $T: X \rightarrow Y$, with $Y$ any Banach space, is compact.

I know if $X$ is reflexive, then any Dunford-Pettis operator is compact. But I don't know how to use it here. Also, is it possible to come up with a solution without using the Rosenthal's $l^1$ theorem?

Any help will be appreciated.