The Planck length might or might not be a physical limit of the universe. We don't have any specific proof that it's the smallest, just that we will not be able to observe any of that size or smaller. To look at something, we need to use light, and we must use a wavelength smaller than the details we wish to resolve. For something of the Planck length or smaller, this ultimately results in a photon that would have more energy in that area than can exist without a black hole forming... so one does, which then prevents us from measuring it, much less anything smaller.
Space and time might very well be discrete and not continuous - certainly the Loop Quantum Gravity folks would agree there. But there are widely supported theories that take both sides.
(I tend to lean towards them being discrete, but I would hesitate to call myself even an amateur hobbyist when it comes to theoretical physics...)