Ehsan, you are right. This is the passé composé tense of a verb that uses avoir in this tense. The participle must agree with the direct object, but only when the direct object precedes the verb.
As you know, the direct object here is 1st person m', referring to the speaker, and it does precede the verb, a. Therefore:
If the speaker is male, Elle m'a mordu.
If the speaker is female, Elle m'a mordue.