Which is the most distant galaxy?
Probably the galaxy known as HD1. By measuring the "redshift" of the light coming from HD1, we see a galaxy that is around 13.5 billion light-years away. Taking into account the rate of expansion of the Universe, this shows that HD1 is actually about 33.4 billion light-years from us (its "proper distance").
HD1 was discovered by Fabio Pacucci and his colleagues at the Harvard-Smithsonian Center for Astrophysics in Massachusetts, USA. They found HD1 by searching data sets from several of the world's most powerful telescopes including the Atacama Large Millimeter Array (ALMA) in Chile.
Another candidate for most distant galaxy is GN-z11, around 13.4 billion light-years old, or 32 billion light-years away in proper distance.
GN-z11 was much closer to us in the distant past. 13.4 billion years ago, it was only 2.66 billion light-years from the Milky Way.