No, you don't have to be non-stop available with a company phone.
[This article reflects the legal aspects in Germany. Laws may be different in your country.]
The Working Hours Act (ArbZG) does not provide a general entitlement to round-the-clock availability, even with a company phone.
In principle, you do not have to be available to your employer outside of working hours, nor do you have to answer e-mails or similar.
Working Hours Act
If you have made other agreements with your employer regarding availability, you must of course adhere to these agreements–within the limits of the Working Hours Act.
The Working Hours Act limits daily working hours to ten hours, weekly working hours to 48 hours and prohibits employment on Sundays.
Exceptions and emergencies
On-call duties are an exception, but these must also be paid separately. In addition, an exception can be made in emergencies.
§ Section 14 ArbZG regulates the handling of emergency situations. The illness of colleagues is just as little part of this as, for example, a late delivery to a customer.
§ Section 14, on the other hand, lists, among other things ...
- the imminent spoilage of raw materials,
- the imminent spoilage of foodstuffs,
- the failure of a work product, and
- work that cannot be postponed for the treatment, care and supervision of persons.
Read more about this: § 14. ArbZG