Back in the days of analog cellphones, Motorola sold one that was very small. You wore it on a cord around your neck.
For the times, it was pretty high tech.
Anyone know the name of the model?
Incidentally, people do manage to sell old ananlog phones on eBay-is there still service available for ananlog phones?
Startac?
The the smaller members of the Motorola MicroTac family are the only truly old school cell phones small enough to go on a neck strap.
The first approved, actual Cell Phone that I recall was the Motorola Dynatac. It was introduced in both portable models, and moble models.
And it was huge. The OP is looking for a small phone.
First clamshell model. Small and light enough that I recall ads where women wore them pinned to their clothes.
Thanks for the link. I guess I had forgotten how expensive these things were! $2500 for a cell phone!
So which one was it Ralph, the mirotac, dynatac or startac? I had a startac once and thought it was totally tubular man. 
Another candidate is the Motorola Vader. I was already carrying a diminutive Startac at the time I first saw one but my reaction was “That thing is tiny.”
Analog cell service is almost gone. FCC allowed wireless providers to start shutting off analog service in 2008. There are certainly still places where you can get an analog signal, but they tend to be rural areas without other service.
FWIW…the first mobile phone was owned by Elvis.
Though carrying around something the size of a suitcase stretches the “mobile” part.
My brother used to have one of the old bag phones. It wasn’t quite as large as a suitcase, but it wasn’t exactly mobile like people think of today.
I see there is one on eBay that some fool is asking $19.98 for.
Bag phones are great because they have a 3 watt transmitter, as opposed to the .6 watts a normal cellphone has. Analog service, but they can get a signal in places no other phone can.