I am interested in a stochastic process on $[0, +\infty)$ which has a.s. Lipschitz continuous sample paths.
Does such a process with independent increments or markovian property exist? Are there any well known such examples?
What about the same question for cadlag, piecewise constant paths?
There are a lot of nice ODEs of the shape $\dot x = f(x)$ which will have Lipschitz continuous paths. Now, since increments are non-probabilistic, they are independent. Furthermore, this process is Markovian. Similar construction applies to cadlag, piecewise constant paths.
If you would like to have non-trivial probability distribution of any increment, then for the latter case compound Poisson processes (which includes standard Poisson process) are Markov processes with independent increments.
At the same time, I do not think that there exist a process with Lipschitz continuous sample paths, whose increments are independent but follow non-Dirac distributions. Unfortunately, I am not sure whether this is true or whether I've seen a proof of this fact.