Simon-
Yes as long as you are not preventing crawling in some other way. Googlebot crawls at a slower rate or at least indexes slower than non secure page.
Here is a great example- https://www.cia.gov/
The CIA site is all https, and is fully indexed. It helps it’s built in Plone which is like crack CMS for Googlebot.
If it looks like you are having problems getting indexed, let me know and I can give you a couple of other tips.
Oh and have an Imperial for me down in CR.
Preferably at Tango India :hattip: