Yes, it is possible wear an helmet, Look up in your national dressage rules the terms
Caldene Ultra and it is literally the best had I've ever had. It's the most vented hat I've ever seen and I mainly hack and school in it (not had it very long so haven't had the chance to do any competitions!) It is so comfortable inside and the lining is washable so no smelly lining! Also the peak is detachable if you did want to wear a silk over it.
I ride in a Dublin Arista helmet, I use it for Dressage and Eventing (I know it has a peak, however the peak is very soft and it is designed to break should I ever be in the position to need it to). it is ventillated, so I don't get too hot and sweaty during summer and I find it really comfortable to ride in.