Can someone else publish a paper based on independently achieving a result that is in a preprint of mine?

Do you know that they were aware of the preprint? Scoops are bound to be common if many groups are working on the same problem or starting to flesh out a new area with lots of low-hanging fruit. If they were aware of your preprint, and you can prove it, they should have cited your preprint at the very least, and you might get them to publish an errata with the citation. But, if they didn't know about it, they got there first, and you'll just have to accept that.

This might depend on field, but it is not uncommon for two papers to appear on the same thing at pretty much the same time, and both might reasonably be published. In my experience, if a preprint appears on something someone else is writing, they will post theirs as soon as possible, to make it (reasonably) clear that their work is independent (preprints are standard in my field, although some papers are published without preprints being available).

If the paper appeared while yours was under review, and no preprint was available before (showing, for example, that they actually got the result two years before you), it is clear to the journal you submitted to that your work is independent. My expectation would be for yours to be treated as it would have been, but then but if accepted you would include a sentence pointing out that the other work was independent (and published while yours was under review, maybe).

If your paper is rejected by the journal you submitted to, you may find it hard to get it published elsewhere, if the time-lag between the two papers becomes too large.

I would say, let the review of your paper go forward. If it is rejected because of the other paper, well that's what happens sometimes. But maybe it will be accepted anyway. The decision is up to the journal editor. I don't know what this paper is about, but in some fields, replication of results is sometimes useful.

If your paper is published, maybe you can add a note to it about the other paper, and saying they were independent. Maybe even casually mention the date your preprint was on-line.