No, position and time are typically continuous variables in quantum mechanics. You can have formulations in which they are discrete but they are not required and are relatively exotic. QM certainly doesn't say they must be discrete.
I feel that a lot of people here are confusing the math and "reality".
You're definitely correct about the math, i.e. the systems that we humans have invented to model reality. But I guess most of us don't really care about what mathematical model scientists like to use (especially not whether they're "exotic" or not), but rather what reality could be like.
And the quantum properties of QM do seem to suggest that there's some sort of fundamental discreteness in reality. And it seems to run contrary to the resolute claims that reality must be continuous as if it were a proven fact. What I understand is that the math most commonly used by scientists is definitely continuous, but whatever we can measure seems to have some kind of planck limitation.
So are we talking about empirical science or science-flavored theology here? Have we actually found empirical evidence or proven the continuousness of space/time?
I responded to a couple of people who claimed with great certainty that QM meant spacetime had to be discrete, when it says nothing of the sort. I haven't claimed we have proof that it is continuous and I doubt we ever will as that seems akin to proving a negative existential.
Your penultimate paragraph suggests some confusion about ideas like Planck scale and quantisation.
Firstly, there is nothing special about the Planck length itself. It's just a unit of length. Around that sort of scale, though, our current theories of physics happen to break down because both quantum and gravitational effects become significant. That doesn't imply spacetime is discrete (or preclude it being discrete) at that scale. It's just a realm that our current theories don't work in.
Secondly, while describing aspects of nature that are quantised was a large part of why quantum mechanics was developed (and the source of its name), it in no sense says anything like "there's some sort of fundamental discreteness in reality". Quantum mechanics deals with both discrete and continuous observables in a single framework: functional analysis, essentially. The set of possible values for an observable is modelled as the spectrum of an operator, which can be either continuous or discrete. Which sort of observable is appropriate for a given physical theory is a choice made in constructing that theory. For things like charge and spin we use discrete (quantised) values because we have evidence that those things are quantised. For things like position we use continuous values and have no evidence that using discrete observables would better match nature.
Space could in reality be either discrete or continuous, or not even exist in any form we'd recognise as "space" on those scales. Quantum mechanics doesn't give us any hints one way or another.
