Of course. How do you expect to connect to HomeKit without internet.
Actually HomeKit "proper" runs just fine without internet. The devices connect via local WiFi or BT, the hub (eg aTV schedules everything locally) there is no need for a remote connection.
Against that
(a) plenty of HomeKit hardware requires its own internet connection. This is especially true if it's of the "Works with Alexa, Google Home, and HomeKit" variety; it doesn't seem to be true for pure HomeKit stuff.
(b) If you want to record your HomeKit video on Apple Servers, then, well sure, of course you need an internet connection!
(c) it's not clear that Apple's current implementation of all this is optimal. There is value in the idea that your home keeps working without an internet connection. BUT the current Apple implementation doesn't seem to have very good handshaking between your phone, iPad, aTV etc. So you get things like slow status updates at times because your phone wants to engage in its own handshake with every device, rather than just asking your Apple TV what's up. (This is especially bad, in my experience, soon after you leave the house and are now on LTE rather than WiFi. I frequently have five minutes of various devices displaying as "No Response" until the iPhone gets its act together and figures out maybe it should talk to the aTV.)
It's hard to tell how much of this general HomeKit irritation is lousy, buggy programming and how much is a flawed design.