DLC doesn’t “install” on PS4 in every situation, in some cases, like with all of Conan Exiles’ DLC, the data for the DLC is in the patch, all the download button does in the Store is unlock the DLC for you.
You have possibly also overlooked the DLC items in the feats section as well, as unlike the others that have different coloured backgrounds in their icons to differentiate them from the standard items, the Arena items have the same black and red background to their icons that the standard items do.
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