Oh.... let's throw a hand grenade in. 'Premise' is a valid word. It means 'a fact from which a conclusion is drawn'. As in: "All mammals are warm-blooded." "Therefore an animal that is not warm-blooded is not a mammal."
A premise and conclusion can be divided into a major premise, minor premise and conclusion. So "All mammals are warm-blooded." (Major Premise) "I am not warm-blooded." (Minor Premise) "Therefore I am not a mammal." (Conclusion). Oh look - two premises there!
But it is likely that there is perfectly labelled 'On-premise software' if the software is deriving facts from single other facts.
I'm not classically educated. But I readed loads of dicshun... dikchu... word books.
Ah... Grasshopper (sorry, young Roy), a trap for young players...
In the 98 million hits that young Neil found, I'll wager
not one was used in the sense you mentioned. They were
almost universally incorrectly using it to mean software residing on the entity's location.
So to shed some light on WHY the term is in the plural when dealing with (at least) buildings etc. In the olden days (yes, before I was born), the contract for sale for land and buildings was comprised - and still is - of a series of clauses which were then known as "the premises". Thus a contract was always comprised of a number (>1) premises. Over time, but still before I was born, the term "the premises" transferred from the contract to the contract
ed,
that is from the piece of parchment to the land and buildings!!! So that's why a building is "the premises" and NOT "the premise".
So, when "lobbing a grenade", make sure you've taken the pin out...

Oh, and don't forget, over time (but after I was born), dictionaries have abrogated their one-time primary role to tell you what a word or phrase meant to merely listing usage. I once (on-air) accused the editor of a well-known Australian Dictionary of "adding to the entropy of the language". Funnily enough, they cut me off. Observations over the ensuing decades have borne out my accusation.
Have a good weekend everyone!
Paolo