'archly' originates from the adjective 'arch' (Middle English 'arch(e)') combined with the adverbial suffix '-ly' (from Old English '-lic', meaning 'like' or 'having the quality of').

'arch' meaning 'mischievous, cunning, or impertinent' appeared in Middle English as 'arch(e)'; the adverbial form 'archly' was produced by adding the suffix '-ly' in Late Middle English and has persisted into modern English.

Initially it meant 'in a wicked, crafty, or impertinent manner', but over time it softened to the current sense of 'playfully or teasingly sly'.