Mistress is precisely the word you're looking for and not archaic or obsolete. US English dictionaries: MW (4a), Wikipedia, TFD (1); British English dictionaries: Collins (1), Cambridge (2), Oxford (2). In English, it doesn't imply that the man is taking care of the woman financially or is significantly older, though it is a common choice (probably the most common) in this situation.
This is the best and most correct choice. It's not specific to a particular dialect, unambiguously defines the situation, and is acceptable in formal registers. However, it's worth noting a few things:
- Mistress doesn't entail secrecy, but there is a connotation of it, and it is usually inferred. Most men wouldn't want their wives finding out that they're sleeping with another woman; this is understood.
- While not outdated or obsolete, it's not a new word or usage, and it's formal. If you're hanging out with your bros and swapping dirt on who you're bangin', you might raise some eyebrows by saying you've got a mistress. Match your diction to the context.
- Mistress is the typical title for a dominatrix, though that's a different definition of mistress. It's very unlikely that the two terms would end up being conflated in conversation, but it's not impossible in the the right context.
Other potential synonyms, and reasons they don't quite fit what you're after:
- Lover - doesn't specify a gender or matrimonial state of any party (though it's definitely unusual to refer to one's spouse as one's lover, it's not incorrect). The right context will make the meaning you want understood: if you tell me you've taken a lover and I know that you're married, then I'll understand. However, lover on its own won't convey the necessary information.
- Girlfriend - a woman the man is seeing and not married to, but this is most commonly used when the man isn't married. Again, context makes a difference; if you introduce her as your girlfriend to someone who knows you're married, this fits the bill. But if the listener doesn't know you're married, it won't.
- Something involving [on the] side (e.g. side action) - slang, not gender specific (without something like woman), and doesn't entail that the man is married (though it does mean that he has some sort of other, primary relationship).
- Paramour - archaic, not gender specific, and doesn't necessarily mean that the man is involved with another woman (married or not). It means that the couple's love is somehow illicit, but there may be other reasons for this than cheating on one's spouse. This does have the advantage of connoting secrecy much more strongly than mistress.
ELU has been asked for the equivalent term for a man sleeping with a married woman, and it seems there really isn't a direct match.