"railroad with 'Branch' in the name" is the way I interpreted the question. Any other interpretation was pure troll. So a quick non-comprehensive search yields (not including a vast variety of model railroads, and mostly not including lines that are simply branches of a bigger railroad)...

* Buckingham Branch (VA): [buckinghambranch.com]
* Central Branch (KS): [centralbranchrailroad.com]
* Saugus Branch (MA; now a trail): [en.wikipedia.org]
* Twin Branch (IN; abandoned): [www.monon.org]
* Short Lines & Branch Lines Vol. 1 (Youtube): [www.youtube.com]
* Eagle Lake and West Branch (ME, long abandoned but now a wilderness trail with rusting abandoned steam locomotives): [www.maine.gov]
* South Branch Valley (WV): [transportation.wv.gov]
* Columbia Branch (MO): [www.american-rails.com]
* NY & Long Branch (NJ, now part of NJT): [en.wikipedia.org]

For "branches" in general, [www.abandonedrails.com] is a great source for those that are no longer with us. Another fun site to explore is [www.openrailwaymap.org]