Is there a smallest time interval?
There is something called Planck time, which in some theories could be the smallest unit of time.
If there was, it wouldn't be Lorentz-invariant. There would be some privileged frame with minimal time quant and we could check whether we move relative to it.
Couldn't you just quantize proper time then, and let coordinate time be anything?
It's not a smallest interval in the sense of a pixel grid for pictures, where you snap from one to the other. As another reply explains, that would not be Lorentz-invariant.
However, the Planck time (about 5.4e-44 s) is a "smallest interval" in the sense that if two events occur that close together you can't meaningfully say that one came first. In quantum physics, you only ask about ranges: does some event occur in this box, with what probability? Will my particle act between t1 and t2? The theory of QM doesn't work smaller than that. Maybe something else does; we don't know yet.
It's not a tick, tick, tick snap of discrete clock points. But it's a resolving power, where you can't tell apart anything that close together.
Sounds to me like there is a smallest time interval then, since anything beyond observation is not significant.
If you're talking about box, you must have some other dimensions involved. Then you can extend the other dimensions and get smaller time granularity - and there is no smallest quant.
If you don't use other dimensions and the "box" is made of time alone, then you get a pixel grid. Translational symmetry may still hold, but Lorentz transformations give you a preferred frame.
Separate names with a comma.