DataScript, Datahike, Datalevin, and XTDB 1.x are open-source. (XTDB 2.x is also open-source but has switched from Datalog to its own query language and SQL.) DataScript, Datalevin, and XTDB have been used in production; not sure about Datahike. All of these databases come from the Clojure community and target Clojure as the primary language. The XTDB team has published a comparison matrix at https://clojurelog.github.io/.

Aside: I write a lot more Python than Clojure, and I wish someone ported Datalevin/Datahike/persistent DataScript to Python. I'd try it as an alternative to SQLite. I suspect with thoughtful API design, an embedded Datalog could feel organic in Python. It might be easier to prototype with than SQLite. There are Datalog and miniKanren implementations for Python, but they are not designed as an on-disk database. PyCozo might be the closest thing that exists. (A sibling comment https://news.ycombinator.com/item?id=40995652 already mentions Cozo.)

Not sure if "production ready" but it's worth looking at Cozo:

https://github.com/cozodb/cozo

Has a dialect of Datalog + some vector support. Multiple storage engines for backend including SQLite, so if your concern is data stability that seems like a reasonable, proven option.

Compiling Datalog to SQL with Logica is possibly the easiest path if you need a production ready open source Datalog setup (i.e. choose your favourite managed Postgres provider): https://logica.dev/

Datomic uses Datalog with a weird clojure syntax instead of the usual prolog-like syntax.

Not open source though?

No. It's only free as in beer. There's some weird mention about the Apache 2 license, but it only applies to the binaries, for some odd reason.

Hmm open source I'm not sure, there are many SQLite equivalents listed on wikipedia though, if that counts.