I can't find info online, but I think that the airside bit of the airport closes after the last departure (also, bear in mind the airport is your classic Ryanair place, so neither humongous nor super comfy). As for sleeping landside... I did it a couple of times in my younger years and it's sh*te. Plus, with Covid, you'd be obliged to wear a mask and it might not be possible anymore - there's a notice on the website, from 2020 but never changed, informing passengers that the airport shuts during the night.

There's a "capsule hotel" at BGY: https://www.zzzleepandgo.com/index.p...ve=MXP&lan1=en

Alternatively there's an NH hotel on the other side of the motorway, connected by an underground walkway (or a shuttle bus).

https://www.nh-hotels.com/hotel/nh-o...YaAgwFEALw_wcB