Unknown. There's at least one approach to quantum gravity based on the idea that it is (causal sets), but we haven't proven it one way or another.

I'd rather ask: Is space discrete? We know there are discrete particles, but AFAIR they can potentially move to any position in the continuum, i.e. for each particle, x, y and z are in the domain of reals (or some huge real interval), not some discrete subset of it.

Or, am I mistaken?

It's the same question (since time and space are not fully distinct), and the answer's equally unknown.