Hello 23475!

You have been around this forum for quite a few years.
I'm sure you know that giving birth usually symbolizes
a new idea/project or direction in your life.

It starts in an office. This makes me wonder if you were offered a project at work which you felt was not appropriate for you due to your age. It was turned over to someone else, but now you are having second thoughts about it.

Then again...

Offices and hospitals are both symbols of the mind. The hospital would also indicate health. The mind can be a powerful influence over your health.
So this dream might be asking you to be more aware of your thoughts. Your thoughts of being too old may be limiting you in more ways than you know.

The baby turns out to be a girl, which would represent you.
She's given to a woman who you see as controlling. This could be someone in your life currently who you feel has too much power over you. It may also represent your mother.

As I see it, this dream is asking you to review your views of yourself. Perhaps you are too old to have a child, but you are not too old to change your mind.