Getting a deportation liability notice doesn’t mean you can be deported straight away–it simply begins the process. You have various appeal rights (depending on what kind of visa you’ve got and why they’re deporting you) that you can use before you’re deported.
Immigration New Zealand should give you a chance to use your appeal rights. They will then hand you a deportation order and force you to leave the country.
After they’ve made a deportation order, you can still ask Immigration New Zealand to cancel it. They have a broad power to do this.