A ten month old can understand if you hand them a teething ring or food and say "We bite this, NOT people" (or mummy, or daddy, or whatever word you know she understands). It worked for my DS. And no, he didn't start biting as a way to get toys, how ridiculous. He knew he could get toys any time by pointing to them and making a noise, or, you know, crawling up to them and getting them himself! Or you can do the no and put them down thing, as others have said.
I am very taken aback at all the pro-smacking posters on this thread, most of the time mumsnet seems very anti-smacking. It is your choice together with your DP of course, OP, but just be aware there are many many alternatives which work, there was a recent thread on AIBU called "To ask how you discipline" or something, which followed on from a smacking debate thread and it was very interesting. You can also look in the Parenting section or post about particular issues there if you want to find out what other people do, usually you end up with a range of options so you can pick which suits you and your family best.
I think you need to have a serious discussion with your DP about this issue, if he is pro smacking and you are strongly anti it, you need to discuss this NOW before she becomes a toddler. I know a lot of people who used smacking with pre-verbal children for example if they were about to reach up to something dangerous - the fact he's using it as a general discipline tool suggests to me he's likely to carry on doing this, and as she gets older, it's going to get harder, so you need to discuss now if you are not happy with this and come up with a solution you are both happy with. You don't have to parent in exactly the same way (DP and I think it can be good for children to see that there are different approaches to things) but you need to be respectful of each others' choices, and back each other up. You can't do this if you feel the other is being completely unfair, so discuss it way before it comes up, to prevent some kind of standoff happening in front of her.