Normally, the two main sources of cosmic rays are considered to be the Sun (solar energetic particles) and galactic or extragalactic sources, although extragalactic sources are more common. Surprisingly, though, there is a third source, a family of particles known as anomalous cosmic rays (see Hovestadt et al. (1973) and Garcia-Munoz et al. (1973)). These are particles - generally ionized nuclei of lighter elements - primarily originating from interstellar space with energies on the order of $\sim10$ to $\sim10^2\text{ MeV}$.
Anomalous cosmic rays are generally accelerated at the termination shock of the heliosphere, moving into the heliosheath. Some may then escape into the inner Solar System. Here's an image from Stone et al. (2005), which shows them as a distinct population from galactic cosmic rays:

Now, these particles are technically of interstellar origin, even though they are accelerated to great speeds only at the boundary of the Solar System. However, a related population may have been found in the Kuiper Belt (see Schwadron et al. (2002)), formed from grains there after collisions between objects. While these are not necessarily - or even likely - the dominant source of anomalous cosmic rays, they do exist, and thus constitute another source of cosmic rays inside the Solar System.