A simple counterexample is a process that, starting at zero moves with constant velocity 1 (with probability $1/2$) or with constant velocity -1 (with probability $1/2$).
There are various subtleties here, and the understanding of what a diffusion process is has varied in time and from one author to another.
In my favorite introductory book on stochastic processes by Wentzell (A Course In The Theory Of Stochastic Processes, originally in Russian), with his choice of definitions, a sufficient condition for a Markov process to be a diffusion is essentially a combination of stochastic continuity and nice behavior of (truncated) mean and variance of transition probabilities.