@terziyski 

Thanks so much for taking the time  to help. I tried what you suggested. My router is at 192.168.1.254 so I set the range extender to be static on 192.168.1.6. I set the DNS as the router address and turned off DHCP.

My router is a BT smart hub 2, which broadcasts on both 2.4ghz and 5ghz, but with the same network name - they can't be disentangled. In case that was causing an issue, I also connected only to the 5ghz band.

I'm going to spell out exactly what happened next in case any of it has any significance:

1. I connected to the extender's network (TP-Link_Extender) from my laptop.

2. I tried to load both tplinkrepeater.net and theatlantic.

3. The network disconnected, and then reconnected to a network called TP-Link_Extender 2 (the original TP-Link Extender network disappeared).

4. both theatlantic and tplinkrepeater.net loaded, but then immediately dropped. And now I'm back to the same issue again. Some sites (like theguardian and google) work. Others (like theatlantic and the actual range extender's web interface) don't.

  @terziyski 

Thanks again. I did try that, and regardless of which band I'm connected to, I get the same issue. Internet randomly drops to 'no internet' or it is connected but seemingly random websites won't load.

It really seems like some kind of DNS issue, or maybe some kind of naming conflict on the network?

I definitely have the latest firmware (I checked). 

  @terziyski 

Thanks for the continued efforts to help. Unfortunately I can't disable the 5ghz band. My router (the BT Smart Hub) doesn't allow disabling individual bands. It's a very annoying limitation of the router.

  @terziyski 

Thanks - but that's not going to work for my setup unfortunately. I've got TP-link tech support calling me today. So if they are able to resolve the issue then I'll post the solution here.