'arena-like' is a compound formed from the noun 'arena' and the suffix '-like'. 'arena' originates from Latin 'harēna' meaning 'sand', and the suffix '-like' comes from Old English 'līc'/'lic' meaning 'having the form of' or 'similar to'.

'arena' entered English via Medieval Latin 'arena' (from Latin 'harēna') where it referred to the sandy floor of an amphitheatre and then the place of public spectacles; the adjectival suffix '-like' developed from Old English 'lic' and later became the productive Modern English suffix '-like'. Combined, they formed the modern compound 'arena-like'.

Originally 'harēna' referred to 'sand' and then to the 'sandy area used for contests'; over time 'arena' came to mean the venue for contests or spectacles, and with '-like' the compound came to mean 'resembling such a venue or its competitive/public character.'