A possible work around could be done via "Save As" or "Save as Copy." Although this is not a perfect solution as it would depend on your artwork set up, so take the time to see if it fits your needs and makes the process easier for you.
Place the artwork you want as separate files onto separate artboards if you haven't already done this.
Then, via Save as or Save a copy, save as an .ai file and click "Save each artboard to separate files."
This will save each artboard as separate illustrator files. It will only keep content associated with the artboard in the separated files. This includes content on the artboard and content partially on the artboard. It also includes content that is grouped but appears off the artboard. This isn't an issue but worth noting, as you can consider this in your artwork set up from the beginning. Which will make life easier in the long term.
You could then link the separate files as you wish and make amends to the newly separated files. Alternatively amend in the single file and just repeat the save process and overwrite the files.
However, you wont be able to have a say in the file names like the Export features offers. You would have to accept having the file name with numbers, i.e [File Name]_1, [File Name]_2 etc. If this isn't an issue then happy days! If it is then you could manually change the names and replace the linked files but then that might just be more hassle if you have a lot of files to rename.
If you want to get fancy you can layer artboards on top of each other and use the same process. Without going into too much detail, I have done this in the past on some bespoke print jobs that require a lot of manual craft work afterwards and I am after particular alignments. I have also done this to pure vector storyboards or artwork I want to split on mass but still want to edit afterwords.
On a final note, look for scripts ;)