Keen students of cryptopolitology will know of the so-called "Fast Cities", places like fluvial Ambulatia and Velocester, which currently holds the urban speed record. But the fastest city ever to have existed is Urbs Ubiquis.
It began as New Exeter, founded on America's east coast by a breakaway group of Mayflower Pilgrims in 1621. The schism between the main, Puritan colonial party and the more liberal splinter group was a bitter one, and out of shame and resentment the governor of New Plymouth later claimed that the New Exonians had never left, but died from an outbreak of scurvy in the colony's first winter.
As for the New Exonians, their anticipation at leaving England behind, compounded by their desire to get away from their fellow Pilgrims, was so strong that the city itself began to migrate towards the north-west. The movement was at first measured in inches, but the city accelerated at what appears from surviving records to have been an exponential rate. This uncanny movement exacerbated the rift between the citizens and their superstitious countrymen, but doesn't seem to have bothered the Native Americans, who were happy to trade with the "running city". Over the course of three years or so, the citizens took on food crops, livestock and fresh citizenry from their neighbours, initially the Wampanoag and later the Pequot. In 1624 New Exeter's population topped 1,000. By this time its speed was such that it could only be approached on horseback, or head-on, although the head-on approach was considered inadvisable.
The city's exact route after this time is unknown, as it sped out of Pequot territory and into America's unoccupied heartland. Many small towns today claim that their valley, or that strip where the grass just won't grow, was scorched by the city as it blazed its trail north-west, but after more than 300 years of colonisation, civil war, expansion and development it's impossible to be certain. What is known is that before it reached the foothills of the Rocky Mountains (and who knows which might have done more damage to the other?), New Exeter's speed went right off the scale, and it hurtled out of topological reality altogether.
It was after this time that the citizens renamed New Exeter "Urbs Ubiquis", as they came to understand that their city was now... everywhere. (Most recently, the younger generation have taken to calling it "Zen".) Spread so thinly, however, Urbs Ubiquis and its inhabitants are not quite as solid as they once were.
Ubiquis's towers and minarets can be seen out of the corner of your eye, in some fortuitous juxtaposition of your own city's buildings perhaps, or by some trick of the light shining off glass or metal. When you glimpse for a second a human face in the fire, or in the pattern of your wallpaper, or in the branches of the tree outside your window, you glimpse the people of Ubiquis about their daily business. It is possible to love an Ubiquinian, or to be loved by one, in the half-felt breath of a warm breeze on your neck, the half-felt caress of your clothes pressing against your ribs. It is possible to hear their voices breaking through on talk radio. It may be that Dali's "Spain", "Invisible Man" and other similar portraits were occasioned by visits from Ubiquinians.