You mention only at the end that she sees a therapist. That seems pretty vital here. If the therapist is talking with her about worry -- does she suffer from excessive anxiety? You mention you don't want to cause her "to feel so darned much anxiety" about growing up, but if she's in therapy working on anxiety already, do you somehow feel therapy is not working, or do you wish it were working faster so she would be less anxious?
I would be talking with her therapist about your own concerns that she is -- what exactly? Not as mature or self-sufficient as you believe she should be at "just turned 11"? Is the issue the occasional whining tone, or the larger issue you see with her not really wanting to grow up yet? There's a gulf between those two things.
I would talk with the therapist about your own expectations and your motivations for wanting her to be more responsible, and what responsibility means to you -- it may mean something very different, and scary, to her, so she regresses into a whine and you're focused on the whine, not the fear behind it. Can you talk with the therapist about what frightens your child about growing up? Have you talked with her directly about whatever fears she may have? She might associate growing up with leaving home, and she cannot yet picture living without you (and/or dad and siblings) there. She may be insecure and feel that growing up means having to make all her own choices; she's not ready for that. She may not be able to picture any roof over her head other than the familiar one she knows now. She may associate growing up with paying bills and stressing about jobs and money--even if you are fine financially, kids pick up on or overhear things we don't ever tell them outright, and she may view her parents as spending an awful lot of time on finances and that scares her.
There could be a lot of reasons she doesn't like to talk about growing up and no longer being your little girl. Talk to her -- not when she's whining, not when you're irritated with her about whining, but at a calm time alone together -- and ask her what SHE thinks growing up means. Don't try to reason her out of any fears or explain how the adult world really is; just listen. It might help you be more patient with her transition. That transition is not a jump but many more years of gradual lessons that can't be rushed.
As for specific solutions: Does the idea of "someday you'll have adult responsibilities" come up, for instance, because like many a child her age, she might sometimes shirk any age-appropriate responsibilities you now give her? The key word there is "age-appropriate." If she is balky about doing the things that a just-past-10-year-old child can be doing (taking out the trash, walking the dog, whatever chores are the right ones for her in your household) then you can start with those basics, working with her on making up her own schedule to do them and telling her how important it is to the whole household that she helps keep things running. Be sure not to give her responsibilities or chores that are so overwhelming (for instance, rather than saying "Clean your room," ask her to tackle just her messy bookcase. One task to do well rather than an overwhelming one that is undefined in her mind, even if you as the adult know what you mean by "Clean your room.")
And praise her a lot when she does things -- she is not so old that she is past wanting your praise, so please don't think that at 11 she should just be getting on with things and not care about acknowledgement. Especially if she's insecure or anxious she does still need to know she's doing OK even on getting the trash out.
Another way to help her could be reassuring her that yes, she will always be your little girl and you will always be her mom. Sometimes kids see the whole "grow up and be responsible and move away to college" thing as equaling "Do things on your own, without mom and dad to back you up." Of course as the adult you know it's not quite like that but it doesn't mean she, at her age, can see things that way.
Be sure that you are not inadvertently watering the worry seed when you remind her she'll grow up. I would just focus on actions, and stop talking in any terms of "you need to do X or Y because someday you will be an adult and have responsibilities." By actions I mean giving her the right responsibilities for her current age and helping her do them well with adequate direction, and giving her praise for them. That helps her build confidence in herself as an 11-year-old person, and that will lead to a more confident teen and adult eventually, but leaping right to reminders that she will be an adult makes her retreat into childishness. When she does whine, drop the topic immediately and then leave the room and do something else. That removes your attention and distracts her.