I'm afraid, it is not a linking verb as you say.
The most common linking verbs are 'be' and its other forms such as am, was, will be and so on. Linking verbs do describe the subject. And, the subject with linking verbs don't do any action.
She is hungry - the linking verb 'is' describes her.
Now, action verbs are different. They show that the subjects do action.
She sings nicely - the action verb sing describes her action.
Since you read here, tell me, what is the verb in your example?
My younger sister played Tiny Tim in the Christmas performance.
So to answer this question is- no, it is not used as a linking verb. It's an action verb.