From midnight last night, New Zealand is now at civil emergency "stage 4", which means all except essential services personnel are supposed to stay isolated at home for about a month.
The official NZ government list of essential services appears to have been finalised and published hastily. Naturally, 'the authorities' consider themselves essential as overnight we've become a police state: police and courts are working through the lockdown, albeit providing limited services, health and immigration/customs services too. What will happen as their workers are or suspect themselves to be infected with coronavirus is unclear at this point. Presumably they have contingency plans, plus controls to limit the spread of infection within police stations, court houses, hospitals, customs halls, mail sorting offices etc. ... but staffing and service problems are entirely possible as the lockdown continues.
Since they aren't entirely self-contained, there's also a second tier of organizations supporting the essential services and here the lines get blurry. For example, police cars need tyres, fuel and servicing.
Today we will be revising our personal list of essential home services in light of the lockdown. More tomorrow.