falselyfalsely

Falsely comes from Latin falsus ‘false’, and goes back to fallō ‘deceive, trick’.

1. Latinc. 75 BCE–200 CEfallō¹‘deceive, trick’
2. Latinc. 75 BCE–200 CEfalsus²‘false’
3. Old Englishc. 450–1150fals²‘falsehood, fraud, counterfeit’
4. Middle Englishc. 1150–1500fals³‘false’
5. Middle Englishc. 1150–1500falsly⁴
6. Englishearliest quotation 1910falsely⁴

1. ¹ Wiktionary, Latin falsus
2. ² Wiktionary, Middle English fals
3. ³ Wiktionary, Middle English falsly
4. ⁴ Wiktionary, English falsely
5. Earliest quotation in Wiktionary’s entry: 1910, PRINCETON'S GRADUATE COLLEGE; Issues a Denial [....]: “This could not be more falsely stated.”.

Earlier history uncertain. See Wiktionary.