Does Switzerland have cities built on mountains?
It depends, if you consider Switzerland all the Alps, stretching from Monaco to Bozen, then yes, and it also has a nice view over the Mediterrenean sea...
Italy has numerous walled cities coming from pre / Etruscan times, that lasted untill late medieval times and the Church reform. This is when most of these Towns have been desacrated, all people burned o stakes, and then some more gruel stuff...
San Marino is one of those Walled towns that resisted the Church massacres, but the Old Papal State lands are full of scheletrons of these ghost towns sat on top of peaks. in most of these places today nature has taken over. You just don't see them. Look for the ancient city of Rama, north-west Italy near France-Swiss border.
Swiss of course had other towns, not so big as Rama, that indeed was a city. It is thought it hosted a population of 100k people at least.
All Megalithic walls and sacred stones circles are spread all over the place as it was guarding the best Alpine pass from Italy to France-Switzerland.
Today Claviere is in France and it's a Sky resort town if you will... back then it was something else... it was part of Rama...
Aosta is in Italy but was considering becoming French or Swiss after WWII. It is on the mountain pass, not exactly on top of a mountain.
There is not so much need today for such fortified living, but I guess some old towns-castles has been converted into something more useful like a Museum like that of Messner. In Austerrich-Switzerland. Sorry for the punctuation.
But yeah, today basically all of the Sky-towns all throughout the Alps arc are basically sat on top of some mountain, and are big enough to be considered cities in some cases.
Zurich is only 408mt over the Sea level, it's on the bottom of the big valley that formed the lake there. Not the best candidate for a city on a mountain. By modern standards it does not have many...
Cuzco is 3.399mt over sea level... There is not much to debate on the Inca at least... we can all agree they check all flags..!
Some unaccesible Island mountains could be settled with modern settlers launched from a military plane...
Of course this means Sea and Air vessels serves for something, and settlers do not walk on the waters...
but would need probably an Air carrier with an helicopter at least...