Learn Russian: Добыча, убегавшая от хищника, спряталась за деревом. - The prey running away from the predator hid behind a tree.

You would write Добыча, которая убегала от хищника, спряталась за деревом. The active past participle убегавшая is a direct, slightly more formal replacement for которая убегала (which was running away). It takes the feminine nominative ending to agree with the subject, Добыча.

Why is убегавшая in the past tense if the English translation just says "running away"?

In English, "running" acts as a tenseless phrase here, but in Russian, participles must explicitly show when their action happened. Because the main action (спряталась) took place in the past, the running also happened in the past. Therefore, Russian requires the past-tense participle убегавшая (was running) rather than the present-tense убегающая (is running).

Could we use the perfective past participle убежавшая here instead?

It would change the meaning and make the sentence a bit illogical. The imperfective убегавшая means the prey was in the process of running away when it found a place to hide. The perfective убежавшая would mean "the prey that had already successfully run away (escaped) hid," which breaks the cause-and-effect logic of hiding from the predator.

As a refresher, why do we use the instrumental case (за деревом) here instead of the accusative (за дерево)?