Even in a backyard telescope, if you look at Saturn, its brightness will wash out all but the very brightest of nearby stars because your eye adjusts for Saturn's brightness, not star brightness. (One nearby "star" you may see is actually the moon Titan.)
The New Horizons probe did pick up star images in Ultima-Thule images, if you look at the raw exposures, but Ultima-Thule is really far away from the sun, beyond Pluto. And it's a dark body, requiring sensitive camera settings. Note that some raw exposures from some probes will show streaks that may be mistaken for stars, but are actually cosmic rays. They are usually digitally removed from final versions.
The boundary where planetary exposures will start to clearly show stars is roughly around Neptune's or Pluto's orbit. But processing may have to remove them because time-lapse and/or multiple exposures may be needed to properly capture dim planets and rocky bodies that far out. Any stars caught would otherwise look smeared or smugdy.
https://www.esa.int/spaceinimages/Images/2007/07/The_constel...
EDIT: A larger version of the image, with an equally huge url: https://m.esa.int/var/esa/storage/images/esa_multimedia/imag...