The VectorFirstAid plugin I mentioned below as a solution operates only with top-level objects. It means if there are objects within a clipping mask that is also outside of an artboard, the plugin will ignore them.

But Sergey Osokin wrote a script, SelectArtboardObjects:
https://github.com/creold/illustrator-scripts/blob/master/md/Select.md#selectartboardobjects
It allows to select objects inside and outside of the current artboard, including those inside of clipping masks. The process is not lightning fast, but it’s better to have a way to do it automatically, than doing it manually.

Hi Egor,

No problem to merge this 2 ideas!
Thanks for your reaction!

Best regards!

I like this a lot.

Lennert, would you mind me merging this request?
Or do you think that keeping it separate from the artboards would serve the cause better?

It would be nice to have the possibility to decide to delete the information (image, paths, text,...) outside the clipping mask. That could make the file size a lot smaller. Now you can see in the attachment that there is a lot of information outside the clipping mask and also in this example outside the artboards. If we, once the layout is completely correct, could delete all the unnecessary information, that would be better in my opinion, because that way you have a smaller file size.
So, I wish there could be a button somewhere to make that decision.
Of course I don't know how feasible this is.

Thanks in advance

@Egor: I'll look it up, thanks for the suggestion. I'd heard of it before but somehow never got around to looking into it.

AlexM, Lance, check out VectorFirstAid plugin by AstuteGraphics.
You can set a checkup there to clear all off-artboard art and run it in a several clicks.
VFA is incredibly powerful when it comes to this mundane operations.

This seems like a work that actions can handle... if only Next/Previous Artboard commands could be recorded. So, alas, only scripts or plugins.
Voted.

I thought I had a suggestion for you using the actions panel:

select > all in artboard
select > inverse
delete

but as it turns out, it won't work with more than one artboard - the command in the select menu is "all on active artboard" but in my action is recorded as "select all in artboard". So, art on the currently active artboard is selected, not "any art on any artboard".

voting for this as it seems like it would be very useful.