OL: "a small hotel or boarding house in France and other European countries."
MW: "a: accommodations especially at a continental European hotel or boardinghouse : ROOM AND BOARD
bor less commonly pensione \ pen(t)-​ˈsyō-​(ˌ)nā  \ [pensione, from Italian] : a hotel or boardinghouse especially in continental Europe"
Collins: "NOUN (in France and some other countries)
1. a relatively cheap boarding house
2.  another name for full board"

I don't think it's actually dated, perhaps a little specifically European though (but then again the word is also used in Japanese with this meaning, ペンション)

@@ -18,0 +19 @@
+<gloss>pension</gloss>

I think it is. I really only know the term from French. It's interesting that almost all the examples in the Wikipedia page are from non-English-speaking countries.

https://en.wikipedia.org/wiki/Pension_(lodging)

is "pension" really dated?

GG5, etc.

Pension is rather dated BrE - best dropped. "boarding house" doesn't fit that well as it often means long-term dwellers.

@@ -15,0 +16,2 @@
+<gloss>guest house</gloss>
+<gloss>private home providing lodging for travelers</gloss>
@@ -17,4 +18,0 @@
-<gloss>boarding house</gloss>
-<gloss>guest house</gloss>
-<gloss>pension</gloss>
-<gloss>private home providing lodging for travelers (travellers)</gloss>

https://en.wikipedia.org/wiki/Ryokan_(inn)#Minshuku
Minshuku (民宿) are a low-budget version of ryokan, roughly equivalent to a British boarding house or a bed and breakfast.

I'm American and I've never heard of this kind of pension. I asked some Australian friends and they hadn't either.

@@ -15,0 +16,3 @@
+<gloss>bed and breakfast</gloss>
+<gloss>boarding house</gloss>
+<gloss>guest house</gloss>
@@ -17,2 +19,0 @@
-<gloss>guesthouse</gloss>
-<gloss>bed and breakfast</gloss>