I have some experience with this. Now we live in a decently-sized house with plenty of storage space, but in the past we’ve lived in some shoebox-sized places with no closets, and had relatives that loved thrift shopping.
OK, first of all, stop accepting hand-me-downs. This can be hard with some people who really, really want to give you things, but it is your first line of defense. If they absolutely insist, take the toys when your kid isn’t around, and then immediately donate them to Goodwill or some other charity.
Second, when people ask what your child wants for Christmas, suggest books and clothing. A lot of people will get her toys anyway, but this will probably at least cut down on the total number. (I mean, you don’t want her to get ONLY books and clothes, but if you can get some of the more prolific gift-givers to replace some of the toys with other things, it will help.)
Third, every year before Christmas, go through and get rid of things your kid doesn’t play with anymore. When they get older, you can enlist their help with this. My kids are very familiar with the “round up toys to donate to charity” end-of-year routine. Broken and damaged stuff obviously just gets pitched, but we often have a nice box of good-condition toys that can be donated.
Very large toys that simply will not fit in the house get stored at a relative’s house. “This will be a toy for playing with at Grandma’s house!” etc. Preferably this will be the house of the relative who gave you the gift. If this isn’t an option, I would seriously return the gift or give it away before devoting serious square footage to it inside my home. I do not understand the mindset of people who know that you live in (for example) a 900-sq-ft house with no closets, and give you an indoor climbing gym for your child’s first birthday. I mean, what? But anyway.
When deluged with gifts, I sometimes will also put some back for birthdays. We have a neighbor who “anonymously” (we know who’s doing it but they’ve never admitted to it) leaves several bags full of toys and clothing at our door every Christmas season. While extremely generous and kind of them, this often leads to a massive toy surplus situation, so sometimes I’ll just take things and store them for later birthday gifts.
This situation is always a little difficult, because it’s hard not to feel guilty about having “my child gets too many gifts!” as a problem, but it IS a problem when you have limited storage space and you don’t want your entire living space to be overrun with toys. I know how you feel.