The device is a satellite tv decoder. It normally connects to an adsl router through a reserved ethernet port on the router. The reserved port does not give out IP addresses to laptops connected to it, only the other ports do. So the router probably assumes the tv decoder has a fixed ip.

How can the static IP of the device be discovered?

Pinging 192.168.1.* from a laptop with a static IP set to 192.168.1.1 (just like the router's IP seen from laptops connected to it) fails.

What are your router and satellite TV decoder ?
It seems to be specific hardware provided together by your ISP.

I guess that won't work, but you can try nmap to scan all your lan on all ports. That should give you all connected devices :

To the best of my knowledge ping will not deal with wildcards, so that is why it was of no help.

How does your decoder connect? Is it wireless, ethernet, USB, smoke signal, or what? The ones that use smoke signals are so smelly - electrical smoke stinks. Anyway, how do you know it is using a reserved port on the router? nmap will certainly list the devices on your LAN, but it the device is not on the LAN it won't be of much help. If it is on the LAN it probably does not use a static ip since it would have been set by the manufacturer and they would not have known what network to use. So your router probably has its DHCP function turned on or there is another device on your LAN which acts as a DHCP server, perhaps a separate firewall. Assuming it is your router, follow the manufacturers instructions to connect to it for setup - probably a web page often at 192.168.0.1 or 192.168.1.1 but not necessarily. Once connected to your router, see if it is acting as a DHCP server. If it is (hopefully) it will have a list of ips it knows about, i.e. the ips it has issued since it was last reset or rebooted. It should show the ip and either the hostname or MAC or both. Process of elimination should identify your decoder.

If that doesn't work, give us some more info and let us know what you have found out - what port, etc., etc. What else is on your network? Do you have a separate firewall, does your router have a firewall and is it enabled, and so on. It may be possible to play some games with the firewall and get some information, too. But that is apt to be a rather lengthy process.

Port=ethernet port. 3 ports to connect laptops and give them IP's, one port reserved for a cable to the tv decoder that appears to not get an IP by DHCP or maybe it does.

Ping was tried to 192.168.1.2, 192.168.1.3, etc. Not 192.168.1.*.

A pfsense system connected to the device and configured to give out IP's has a feature called something like ARP that shows the MAC address of the device. Additionally, if I'm reading it correctly, it shows 192.168.1.2 as the IP given to it by pfsense's DHCP server. But ping from pfsense to this IP does not succeed. How do you do the equivalent in linux?

Need more information...
On the TV decoder device, do you have a login interface that can you can login to and view/set IP?
What brand is your router and the TV decoder? Do you have link lights on them?

I suggest to unplug all devices to your router except the TV Decoder and a laptop. Then run nmap command recommended by lougavulin.
Perhaps, it would be better to disconnect all wireless devices also while you are doing this nmap scan.

There must be a login interface to the TV decoder but just typing http://192.168.1.2 as the url on the laptop's browser fails just like ping fails. Even though arp shows MAC and IP=192.168.1.2. The router is on and fully functional, you can connect to the internet from a laptop connected to it. Next I'll try nmap 192.168.1.0/24. Maybe they've put the web interface on a port other than 80. Maybe they're filtering out anything that does not come from the router's MAC at the reserved ethernet port and MAC spoofing is needed.

The decoder might not be configured to respond to pings.

Maybe. There is no hard rule that an embedded device must have a built in web, telnet or ssh server. If there is then it might need to be enabled using the decoder's on screen setup page(s) assuming there is one. I would also expect the on screen setup to show how its ethernet port is configured. In addition access to the decoder via ethernet might be by some type of program or app using a proprietary protocol.

I would also expect the decoder's manual to include some information on its network setup but these days documentation can be sparse.

The ip of that device can be configured somehow - or at least should be documented, otherwise it cannot be connected to anywhere (if it has a static ip). Or probably you have a special router which can communicate with it.
Anyway, at home my set-top-box is connected to a different network, so cannot be seen from my home network at all. If your case is similar you will never find it.

Can't you plug a cable from your home ethernet network to the different network of the set-top-box using a hub or something and assign whatever static ip does not conflict with anything?

You have already found the answer to your question from what I can see. I think you are under some misapprehensions, though. As MichaelK said, the device does not have to respond to pings, many don't, and there is no requirement that says the user has to be able to log on to the device. Everything may be automatic and probably is. In addition, if you have a device providing an address for it there is absolutely no requirement that it have a static address. There obviously is some way for other devices to communicate with it but that may well be through the protocol it uses or it may use a standard port for its type of device. A word about ports, too. You said it was on a dedicated port when actually you meant it was connected to your router by ethernet. A dedicated port normally refers to to a port number, a connection point. There are 65000+ 'ports' that can be used for traffic between computers. Port 80, for example, is used for web traffic. Port 22 is used for ssh connections. Port 445 is used for secure web connections. The port you referred to is a physical socket and is not dedicated. the fact that there is only one thing on it at the moment is totally happenstance. It could just as easily be connected through a hub or switch (or several) and thereby connected to all the devices on the network. Ethernet is a means of being a part of a network not necessarily a means of connecting to only one device.

So, if you have the ip address you were looking for is your question solved (please mark it so) or what is your real question?

I see another post was made while I was typing mine.

A 'network' is designed to keep communications separate. In order to be connected to two networks as you asked you would need to use a device acting as a bridge. You already have one - your router. It has a presence on your LAN and on your ISP's network and transfers traffic between the two. In theory if your device is on another network you could get another bridge and put your laptop on both networks but I don't know that you would accomplish anything unless you know a whole lot more about the device.

Ethernet, by the way, is not a network. It is method of communication just like wireless. That is why you can have both ethernet and wireless devices on a LAN.

And where did I say the device has to respond to pings? Ping is just the first thing you try, doesn't hurt to mention (lest someone says "have you tried ping?"). Likewise, no one said a login page exists, just guessing that there must exist something. No one said you have a device providing an address for the tv decoder, what the router does to the tv decoder is open for investigation. "Assume a static ip of 192.168.1.something" is only a guess because the tv decoder's wire does not connect to any of the PHYSICAL ports meant for laptops etc but there is a reserved PHYSICAL port labeled "tv decoder". Laptops do not work plugged to this reserved port.

What is not physical is the TCP port we are trying to discover, where any unknown process might be listening on in the tv decoder, if any. This is to be discovered, if it exists, not looked up in a manual because there is no manual, only instructions to wire things up.

But I'm not even sure what the IP is yet or how to discover what IP was given by DHCP. How do you discover the IP?

All the home wireless routers I have used over the years, they all had a way to show me "attached devices". Do you have this?
Again, it will be easier to disconnect all devices except a single device to login to router and the TV Decoder.

The decoder might show up on the router's attached devices list if it has that capability. Without knowing anything about the router or the decoder or how they "work together" all I can do is provide some logical guesses.

Need some time cause the hardware is not at my home but at a friend's.