'indismissibility' originates from Latin-rooted English formation: the prefix 'in-' (meaning 'not') + 'dismiss' + the nominalizing suffix '-ibility' (from Latin '-ibilitas').

'dismiss' comes from Old French and Anglo-Norman forms (from Latin 'dimittere' where 'di-/dis-' meant 'apart' and 'mittere' meant 'to send'). The English adjective 'dismissible' (able to be dismissed) produced the noun-forming suffix '-ibility', and adding the negative prefix 'in-' yielded 'indismissibility' in modern English formation.

Initially the components meant 'not' + 'to send away' + 'state/quality of being able', and over time this produced the modern sense 'the state of being not able to be dismissed/removed'.