With "to be" and "to become" unless it's a single adjective.
I don't think that's true. My family is from Western Ukraine and I studied there - this situation would simply be nominative.
For Ukrainian, the Instrumental Case would be used when a certain object is being used (hence "instrument"), becoming something, was something, or certain prepositions (like "before" and many others).
In general, the instrumental case shows an item's role in how an action is performed.
I hit the vase with my elbow.
If English had overt case marking, we would see "the vase" in the accusative because it is the direct recipient of the action "hit", and "my elbow" would be in the instrumental because it is the tool/instrument with which the vase was hit.
(English has vestigial case marking with personal pronouns, distinguishing between "I, he, she, we, they, who" for the subject (nominative) and "we, him, her, us, them, whom" for the undifferentiated object (oblique).)
For whatever reason, Polish uses the instrumental case after "to be" and "to become", even though most other languages use the nominative after copulae because they're just subject complements.
A very good explanation. Russian does this with verbs of becoming (it does not use a copula verb in the present tense), and it was once described to me by my Russian teacher as having a sort of adverbial feeling, a sense of the way one was becoming, rather than an end goal.