I couldn't find a US example, but it certainly happens internationally. EK 412 flies from Dubai to Sydney to Christchurch and the Sydney to Christchurch leg happens the next day.
But - what happens if the X->Y leg starts at, say, 23:00, and the Y->Z leg starts at 02:00 the next day?
Nothing much. They just fly as usual.
Is the same flight number used?
Is it then the case that flight 123 goes, every day, from Y to Z and later in the day from X to Y?
Yes . You can type EK412 into seatguru and for any date you get SYD->CHC at 7:50 in the morning at DXB to SYD at 10:15 in the morning.
I just recalled a domestic example. I was on a "direct" flight from Boston to Las Vegas. Turns out United defines "direct" as "both legs having the same flight number". It does NOT mean non=stop. It also does NOT mean it's the same plane, the same crew, the same gate or that it's a guaranteed connection. It's just a marketing ruse: they simply combine two random legs, give them the same flight number and advertise as the connection as "direct".
This particular itinerary had a stop in Denver. The first leg was delayed, and the second leg departed on time while the first leg was still in the air. So, yes, two identical flight numbers in the air at the same time, probably in the same air controller space. And I had to spend the night it Denver!