Question

Is there a way to show a control on a page ONLY if the page was requested over the local network?

Answer 1

Yes, use Request.IsLocal property, it gets a value indicating whether the request is from the local computer. E.g.

theControl.Visible = Request.IsLocal;

EDIT: the answer I provided above is works for a local host only:

To fully answer your question, the subnet mask for the network should be known.
You also may use IP address extension methods from here: IP Address Calculations with C# (Subnetmasks, Networks, …) and use it like:

var ip1 = IPAddress.Parse("192.168.0.1");
var ip2 = IPAddress.Parse(Request.UserHostAddress);
var mask = IPAddress.Parse("255.255.255.0");
bool inSameNet = ip1.IsInSameSubnet(ip2, mask);

To get the the user host address you could use HttpRequest.UserHostAddress method; but remember host address can be spoofed.