Could milk be making DD2 cough?

[Helgand]

DD2 is 14mo and has been on cow's milk since 12mo. In the last month she has developed a cough (and vomiting at times) which is not an infection or anything the gp can figure out. DH and I both think it is connected to drinking milk. DD1 had 18mths of prescription formula due to intolerance of milk and DH and I and my sis all find milk hard to digest.
We have tried goats milk after a period of water (coughing stopped but DD2 v miserable and HUNGRY!)and the cough came back. GP put her on antibiotics - don't seem to have helped.
Today I am going to try and find some rice milk. Any thoughs/experiences?

try googling 'milk mucus production'

Rice milk is great

Recently stopped my dd drinking milk, which dh loved giving her because in ever increasing amounts because she got an ear infection! Her first one at 3 years old. Things improved as soon as the milk drinks stopped.