Hi; I have z visa and our daughter has s1. Can I get s1 for my partner (unmarried)?

Lee, it is difficult to apply for an S1 type for your partner because you two haven’t got married. You can have a try anyway.
You can provide your daughter’s birth certificate to prove the relationship between you and your partner. But I cannot assure that.
The most secure way is to get married first and then apply for the S1 type for her.