There are many that have adopted machine pistols for various uses, particularly special forces, VIP protection, and for the roles currently filled by PDWs, which means that common troops sometimes were issued them.
The Glock 18 is a selective-fire variant of the Glock 17, developed at the request of the Austrian counter-terrorist unit EKO Cobra, and as a way to internally test Glock components under high strain conditions.
What is NOT a meme, and is useful, but counts as fully automatic, is three or five round burst mode.
Chainguns are a type of autocannon. Heavy, light and general purpose machine guns, like the M2 Browning, M249 SAW, Maxim, Been Gun, MG3, etc... none of these are chain guns. They aren't even all belt fed, which I guess is what you meant to say because I think that's the videogame midwit lingo. They are nonetheless universally accepted as being very valuable in combat.
AIUI, chainguns are any externally electrically-actuated automatic weapons, including both single-barrelled and rotary designs, some of which are autocannons, and some of which are lighter (machine guns).
But, yeah, it makes more sense if GP intended to refer to belt-fed weapons, not actual chainguns.
Most recently, it was Russia dropping 2-round burst on AK-12 after experience with it in Ukraine. M4 is another famous historical example.
someguydave is correct. Compact automatic weapons make sense for highly trained body guards protecting VIPs when discretion is considered important.
Compact automatic weapons still usually have either a stock (even the smallest Uzis do), or some other way to stabilize the gun while firing - e.g. the sling is used for this purpose with some MP5K variants.
As int_19h points out, there are special-purpose weapons made for this (see "personal defense weapons") and they are likely what pros carry.
Additionally, the very long history of machine pistols would indicate the form-factor is a poor fit for the application of any full-auto fire.
This is the primary reason that personal defense weapons (PDWs) were developed in the first place.