Several studies I've read a while ago indicated that it's the internet network which is most resistant to all kinds of disasters. That is not to say that it will stay up no matter what, just that the losses in the infrastructure will be the least of all communication methods and will be easiest and fastest to have it back up.
The Last Ship, for example, featured communications via network of mobile phones with bluetooth (not sure if it wasn't wifi too) to send messages over state lines even. Not much of an example to address your question, but illustrates possibilities.
However, as Gallifreyan says in a comment, there must be someone to maintain the network, and even without that damage would be extensive. I did not watch the series beyond first season (I find all zombie things boring after reading "The Black Tide" series by John Ringo... or stupidly ridiculous like the iBrain or something), but I find remarkably low damage level in it. There should be a lot of buildings burned down, for example, burned cars, destroyed infrastructure, including relay stations and exchange cabs etc. And by a lot I mean at least 50%.
And, of course, satellite network is now sufficiently sophisticated to allow for internet connection. I'm surprised no one really uses satellite phones - you can buy an iPhone add-on to convert it to sat phone for less than $200 (i think), and having a smartphone is half-way-there to the IoT.