I am waiting for Quantum Gravity to become a well developed theory, but, before I get into that...
Ask dffierent physicist to define what time is, you will get various answers and not all of them will agree. Until we have general agreement on what is time, we can't really say if it is discrete.
I would say, in all likelihood, it is quantized -- trouble is, in quantum mechanics, time doesn't exist -- ie: not a variable in the equations used in quantum mechanics developed so far. There is a before and there is an after, that is it. Before the wave form collapses, and after the wave form collapses.
If you accept that, that may lead to the conclusion that time ticks and the collapse is the ticking. But then next question is, which wave form? Yours or mine or others? So time depends on where you are and what you are doing (acceleration for example). What if you are observing me? or I am observing you? or no one is observing?
If you are looking at your watch on your wrist with your arm fully stretched, you are not seeing the current time on your watch. You are seeing the time on your watch between 1ns to 2ns ago, depending on how long your arm is. For you, the photon carried the information of time from your watch to your eyes took 1ns to 2ns to get to you -- in your time. Yet (current accepted theory is) the photon didn't experience any time passage. For it, it is the exact moment of it's creation. In fact, it is the same moment for any photon between it's creation and it's death (absorb/decade) even if these two events happened light years apart. It ticked for you, but it didn't tick for at least one something else that is the photon.
May be someone already have a paper (that resolves all these unknowns) written but not yet publish... If not, I am waiting for Quantum Gravity. When that is well developed, knowledge of time would/may have improved enough to gain some clarity there. However, my gut feel is, we are in deceleration as far as pace of science advancement goes. Theoretical Particle Physics hasn't done much in the past half century. String theory didn't turn out to be the great leap forward that many hoped it was... So on, and so on. And that was when science was a more attractive field. Now, as college graduates majored in "xyz studies" out numbered majors in Physics, Math, and Engineering, our capability of maintaining a technological society becomes questionable. There is a fair chance we would go back to horse and buggies. So, "what is time", along with many science mysteries, may forever be unknown to human kind.