Is stuck a linking verb

The easy way to recognize a linking verb is that a linking verb acts as an equals sign, the object is a different form of the subject (Mary is my sister. Mary=sister); or the subject becomes the object (My feet got wet. feet->wet).

The verb stayed (past tense of to stay) can be a linking verb, for example:

You have stayed the same as I remember you. (You -> same)

The cat stayed at the window all morning. (this is not a linking verb, the window is not another form of the cat)