There is hope. I have the smallest hands ever imaginable (like seriously.... they are absolutely tiny) and I think I play both piano and guitar okay (take a look at my YouTube channel if you want to see how I play.) Having small hands may make it difficult to reach large distances on both piano and guitar but this is only one aspect of playing these instruments and there are certainly ways around it. Most of it comes down to finger agility and strength. When I first started playing, like you I thought there was no hope because I couldn't play the chords and I blamed it on my small hands. But as I practiced and played more my hands became stronger and more agile and I can play almost anything now. On guitar of course I won't be able to reach really large distances from fret to fret but this is rarely needed and with practice I can play almost every kind of chord formation. The same goes for piano. I won't be able to reach any distance larger than an octave but this technique isn't mandatory. If I can't reach a large distance I'll play some notes in my left and some in my right.
So, I wouldn't blame it on having small hands because I am proof that you can get around this issue! You just need to practice practice practice until your fingers build up the strength and agility to be able to play these chords. Try doing finger exercises regularly to help. I promise if you try hard you will get there eventually
I have come to found that there are people with much bigger hands than me who have trouble playing a G chord without hitting the notes in between and it's because it's got nothing to do with hand size, but rather strength and composition of your muscles. Like SATCHMO said, I would give it time and see how you go and how your hands adjust to the instrument!
Sorry for rambling... but I do know how you feel as I went through the same thing!