My DD was the same - until we learned to swaddle her, she spent the first 2 or 3 weeks of her life sleeping only ON me at night. I remember the first night at home after coming back from hospital, blithely putting in her cot, thinking, there you are, this is your bed. Ho ho ho. She hated the buggy, hated the car seat, couldn't lie in her Moses basket/baby bouncer for more than a few mins (on a good day) without crying. Basically, she wanted physical contact at all times. I carried her in a sling most of the day, most days, until she got so heavy I was getting back problems. I can't remember when she got less clingy - when I went back to work at 10 months, I think, and I was lucky enough to find a nanny who was the ONLY person apart from me and DH who she was happy to hold her (she wasn't even happy to be held by my mum, who she saw at least once a week, often more). Finding this brilliant nanny was a complete godsend, actually. Before then I'd often felt as though DD was still physically attached to me, and though sometimes it was lovely and snuggly, sometimes - quite honestly - it was completely suffocating.
Like your DD, she would only nap on me, either in the sling or on my lap after BF. At 9 months we went to a sleep consultant about her appalling night-time sleeping (or lack thereof) and part of the plan involved getting her to nap in her buggy (also essential at that point as I couldn't realistically expect our nanny to carry her around in a sling for an hour twice a day!) This we did by putting her in the buggy and pushing her until she went to sleep - she wailed the first few times, but actually settled down far more quickly than I'd been expecting. She rarely went to sleep silently and immediately, but generally she'd fall asleep after a couple of minutes of low-level crying. Your DD is that much younger and you might not want/be able to let her cry at all, but by that time I desperately needed a physical break for a bit of time each day, and it was a real sanity-saver.
Anyway, now she is nearly 3 and super-independent, wants to do everything on her own, always yearning to do a bit more than she realistically can. She is super sociable, loves other children, and is confident and chatty. She is clingy only in the sense that if I'm around, she always wants me to join in her games/be involved in whatever she's doing, but I think that's pretty normal for her age. Hang in there - I know it's tough, but she won't be like this forever, I promise!