With a perfect shot, sure, however you need to keep in mind that with a .17 you are shooting an extremely light weight projectile, that is made to fragment in small lightly built animals (varmints) as opposed to penetrate deep into the vitals of an animal that is more solidly built.
This means there is a strong potential that the small little fragile .17 projectile will not penetrate deep enough into a comparatively large and tough animal (pig or goat). If the wind blows your little tiny projectile into the shoulder of a pig or a goat, the projectile will probably explode near the surface creating a shallow crater, no penetration into the vitals, and probably a less than clean kill or a lost/severely wounded animal.
Its possible, but it only allows a very small margin for error.
If you are sure you cant go any bigger than a rimfire, then a .22magnum would be MUCH more appropriate for the larger animals (pigs / goats). It'll shoot a projectile that is up to 3 times the weight of the .17, that wont explode on impact, and will have the necessary qualities needed to hit bone and still make it to the vitals.
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