Ability to take extended mags (check, though capacity out of the box is good)
Solid build (check, good rep, used by Turkish military)
Reliable bang (check)
Not much larger than PPK (eh, maybe not. Depends on your definition of “not much larger”)
This is a striker-fired pistol, but still offers the decocker and DA/SA, which is a little more rare among striker-fireds. The out of the box triggers on Caniks are regarded as very good. Despite their origins as a service pistol for the Turkish military, they have carved out a market as an affordable competition gun that allows you to compete on a level playing field against the custom race guns that run up the cost. The barrel length on the more service oriented TP9DA model is 4.07", so just a hair over your suggested barrel length.
The Beretta PX4 Storm series appears to cover all of the bases too, with more options on size and certain other features. Unlike the Springfield XDE, which is discontinued, the PX4 Storm is still manufactured: Beretta Px4 Storm Series
Though he does mention an interesting option where you could put a third party accessory called a “striker control device” that you can use to modify a Glock and give it an extra measure of control and safety so it’s closer to offering what fans of DA pistols are interested in: The Frustrating Search For A Double Action Carry Pistol - YouTube
Once you get into DA/SA semi’s in 9mm, you’re mostly going to have SC versions of full size pistols. These are noticably larger than a ppk.
My first suggestion would be the Beretta PX4 subcompact. The other would be a CZ. I’ve only fired a PX4 once, and it was a full size. The downside for me is that I don’t like something that overall size with anything less than a 3.5" barrel (the SC PX4 has a 3" barrel).
I’m getting closer to finding one that works for me and I like the Canik TP9 Series after seeing the SF and Mete versions in real life and watching the MAC torture test and seeing it beat out a VP9. I really like the DA series and while it does not have the safety like the SF which is more of a trigger disable I think the de-cocker to DA works. MAC Torture Test: TP9 Torture Test! How does it stack up to the VP9? - YouTube and the Mete by Hicock45: Canik Mete SFX - YouTube which I don’t like since it lacks the de-cocker and it has no DA.