Hello @Family and welcome to our community!
Broadly speaking it is allowed that asylum seekers and German citizens get married.
We had a similar question here on the platform quite recently. Please check out the following link:
In your case it is of special importance that your asylum application has already been rejected. That increases the time pressure, especially since German authorities will assume that you are only trying to get married in order to avoid deportation. They will probably start investigating if you and your girlfriend are actually a couple. But based on your description of your relationship I'm optimistic that this will result in a positive outcome (e.g. because you are living together).
Nevertheless, you have to hand in a couple of documents in oder to get married (see the linked thread above) which might take some time if you don't have everything at hand right now.
Keep in mind that giving your passport to the Standesamt could lead to another severe problem: That is, that you can only be deported if you have valid identification papers. If you give it to the Standesamt then German authorities know that you have a passport/ID/etc. That's risky since your deportation is imminent.
Do you have an appeal option for your asylum application or is it an unchallengeable decision? Doing a so called Folgeantrag might delay your deportation because they have to check your application again. That might give you the time you need to get married.
Here is two other threads circling around similar questions that might be of interest for you too:
I really hope this answer is helpful to you. If you have further questions, feel free to ask them here in the community.