IPA: /ˌʌn.rɪˈpɛn.tənt.li/
KK: /ʌnˈrɛpɪntəntli/
In a manner that shows no regret or remorse for one's actions.
He continued to act unrepentantly, refusing to apologize for his mistakes.
Unrepentantly → It is formed from "un-" (meaning not), "repent" (from Latin "paenitere", meaning to feel regret), and "-ly" (meaning in a manner). The word describes a manner of acting without feeling regret or remorse.
Think of 'not' ('un-') feeling 'regret' ('repent') — that's why unrepentantly means acting without regret.