This is a rather emotive topic. Arms are ‘personal’ i.e. they belong to an individual and do not belong to ‘families’ ‘septs’ or ‘clans’ despite what on-line businesses assert when selling copies of ‘coats of arms’ to gullible individuals.
The Barnwell /Barnwall (and associated spellings) surname possibly derives from a Barnwell who arrived in Ireland with the Norman invasion. It could relate to a Barnwell who arrived in Ireland during a later period, but essentially the family is of Norman origin and not ‘Irish’ (Celtic). An early Barnwell might have been armigerous (i.e. have been granted arms) but Irish chieftains did not have arms until at least the era of ‘Surrender & Re-grant’ when, under Henry VIII and Elizabeth I, many Irish chieftains were granted titles and with them came a grant of arms. Both titles and arms were hereditary, so under the English law of primogeniture the senior descendants in the male line of the grantees had the right to bear them. Most of the heraldic souvenirs sold to tourists is inaccurate rubbish.
Irish sept or tribal armies did carry banners into battle but they tended to be a plain colour or in exceptional cases they depicted a sept icon for e.g. the red hand emblem of the O’Neills of Ulster.