'arctically' ultimately derives from Greek, specifically the adjective 'arktikos' meaning 'of the bear' (from 'arktos', 'bear'), used to name the northern regions associated with the constellations Ursa Major/Minor.

'arktikos' passed into Latin as 'arcticus' and then into Middle English (variants like 'artike'/'arktik'), eventually giving modern English 'arctic' and the adverbial form 'arctically' by adding the adverbial suffix '-ally'/'-ly'.

Originally it meant 'of the bear' (i.e., northern, relating to the constellations near the North) and came to mean 'northern' or 'relating to the Arctic region'; the adverbial form later acquired the sense 'in a very cold manner'.