Is my prayer valid despite my clothes being defiled by the saliva of a dog, which I cannot change until I reach home, and there is no time available for changing, given my attempt to avoid the dog and the difficulty of speaking to its owner about the matter?

If a dog licks clothing, the licked area becomes impure, and it must be washed seven times, one of which should be with earth or an alternative. The same applies if the dog sniffs the clothing and moisture from its nose transfers to it, or if its urine or excrement touches the clothing. It is not necessary to change the garment; rather, it suffices to wash the impure area. You can purify it and pray in it. If you are unable to wash it, and you have no other pure garment, and you fear the prayer time will expire, then pray in your impure garment; there is no need for re-performing the prayer according to the Hanafis, while re-performance is recommended according to the Malikis as long as the prayer time remains. However, if you pray in an impure garment knowingly and are able to purify it or have a pure garment, your prayer is invalid and must be re-performed.