> Beyond speculation, I'm not aware of anything showing spacetime to be discrete.

It is a prediction of loop quantum gravity [1]:

In 1988, Carlo Rovelli, Lee Smolin, and Abhay Ashtekar introduced a theory of quantum gravity called loop quantum gravity. In 1995, Rovelli and Smolin obtained a basis of states of quantum gravity, labelled by Penrose's spin networks, and using this basis they were able to show that the theory predicts that area and volume are quantized. This result indicates the existence of a discrete structure of space at very small scale.

[1] https://en.wikipedia.org/wiki/Carlo_Rovelli#Loop_quantum_gra...