I am installing 3/4" x 2 1/4" red oak tongue and groove solid hardwood as a first-timer and just floor nailed a piece in that I discovered has a hairline split in it near the edge. And I was doing so well! Is it possible to somehow remove this piece with a pry bar or similar and replace, or should I glue and secure the next piece? I'm not sure I can get the piece out without damaging pieces to either side and behind the slightly damaged piece. Thanks in advance for all insights.

Is the split on the surface or the tongue? You can remove the piece by cutting the nails with an oscillating multi tool. Use a metal cutting plunge blade. Rest the blade on the subfloor and work it under the hardwood to cut the nail. There'll be a piece of the cut nail sticking out of the floor....pull it out or drive it below the surface of the subfloor. I assume your hardwood has tongue and groove ends? You probably won't have a replacement the same length as the piece you remove so you'll have to cut a longer piece to the same length and add a tongue or groove to one of the ends. What have you got for cutting tools? Table saw? Router?
If the split is on the surface you can rub in a wood filler or putty. If it's the tongue you can glue it....a small split isn't a big concern....install nails on either side of it.

Another option for a hairline split on the surface is to put some wood glue into the split and then put some sanding dust from the same hardwood into the split.