This is very complicated.
One could argue- rightly I think- that the only guns Browning ever made were from Utah and that basically ended when Winchester bought the rights to the single-shot rifle in 1880something.
The great man sold licenses to manufacture his designs to Winchester, Colt, Remington and Fabrique Nationale (now FNH).
Browning developed a very strong connection with FN when they began to make his .32 pistol and typically spent six months a year in Herstal for the latter part of his life.
In the 1920s Browning gave FN the right to use his name and it was widely assumed that there was a Browning Company somewhere but in fact all the guns were made by FN in Herstal, Belgium.
The various P-35 models- including the more recent stuff (BDA) have been made by FN or licensees- hither and yon.
But the key point is other than the Browning Brothers shop there isn't a plant somewhere with "Browning" over the door. The closest thing would be the Browning Custom Shop in Herstal, but it is technically a subsidiary of FNH.
FNH is actually a very big outfit with plants in Belgium, Portugal, Japan and the U.S. They own USRAC (Winchester long guns), FNMI in Columbia, SC and Browning USA- which is more of a distribution company).
It really is complicated.