I've worked it out. It was the sword in the stone by T. H. White. I must have been five or six, and it was a hardcover book; I remember it very heavy.
That is, of course, if you don't count the house at Pooh corner as fantasy. I already officially owned this when I first went to school at four, although I needed help on long words like heffalump.
Narnia came slightly later.
Oh, and Alice? No, Alice came from the library, while the sword was a family book, that just sort of gravitated into my shelf…