The news has come out of their Support page, using a FAQs section to explain any questions fans may have:
Q: Will any future DLC map packs come out for PlayStation 3?
We do not have plans for any DLC to follow Awakening on PlayStation 3.
Q: Is this the end of Zombies content on PlayStation 3, with no continuation of the story?
Yes, there are no plans for future DLC content on PlayStation 3.
This also means there will of course be no season pass for the last-gen consoles so acquiring the Awakenening DLC, which consists of the four maps Gauntlet, Splash, Rise and Skyjacked as well as the Zombies mission Der Eisendrache, will be a stand-alone purchase of $14.99. The pack will become available on April 5th.
This weekend will see the DLC free for everyone on Playstation 4. The Black Ops 3 Awakening DLC Free Trial is coupled with a double XP and double weapon XP event all weekend so there’s plenty of reason to jump back on this weekend even if you already own the Awakening DLC. The double XP event will be taking place across all maps and on all platforms, including last-gen consoles.
It seems however that Call of Duty’s days on last-gen consoles could be coming to an end. The support page linked above also explains the following,
“The Call of Duty: Black Ops III PlayStation 3 and Xbox 360 versions feature Multiplayer and Zombies gameplay only, but do not include a co-op or single-player campaign game mode. The ambitious scope of the 1-4 player co-op Campaign design of the PlayStation 4, Xbox One, and PC versions could not be faithfully recreated on old generation hardware.”
As more work is put into future current-gen titles it’s possible that Xbox 360 and PS3 versions may be cut altogether in years to come.