Here uncontrolled fields are also common but you'd still call local traffic. And transponders are always mandatory. But aviation is not as commonplace as in the US.
Disclaimer: I'm not familiar with Lompoc. That looks like class echo airspace in which a radio is not required. Apparently the airport's control tower only operates part time. Still, with nearby population centers, restricted air over military bases, possible gliders and skydivers in the area, even oil wells relatively close, it looks like a particularly foolish place to turn an aircraft into an unguided missile by jumping out of it.
Sources: https://aeronav.faa.gov/visual/12-02-2021/PDFs/Los_Angeles.p... which includes a legend, faster version here: https://skyvector.com/?ll=34.665619444,-120.467502778&chart=....
And notifying local traffic (on CTAF) at uncontrolled airports is recommended, but not required.
This is quite common in the US, especially in small towns/rural areas.
Also worth noting that in a life-threatening emergency you have wide, wide discretion when it comes to putting her down. Fields, Roads, Riverbed, pretty much anything that's remotely flat is fair game.
We were just philosophising whether he had one or not.