Very much depends on their reasons for leaving. If it's money, sure, though that ties in to why didn't you respect them enough to pay them market ASAP, if you're able to once you know they're trying to leave.
If it's something other than money, chances are real good you won't be able to keep them.
Small, rapidly growing companies are actually the best places for meaningful career growth, getting a 'organic' experience of growing together with the changing roles.
If in your smaller organization really the best mobility is to jump ship, then you have to factor that already in hiring that you need someone who would like an organization that stays small, and beware of ambitious key people who'll not be good for you. But for startups it shouldn't be so, unless you fail.
It's true. Sometimes, until you get big enough, you can't provide a career path in the short term. But you can at least make someone that is truly great a lead, and let them know as you grow, you want to find a bigger path for them. At least, you can tell them you want to do this. Sometimes, they will need to go to achieve their goals if you don't grow fast enough.
But, if you grow fast enough ... this resolves itself.
I've had coworkers who didn't particularly care about money, and were clearly on their way out. In order to retain them, they were transferred to different teams where the work was more engaging.
If it's something other than money, chances are real good you won't be able to keep them.