Air carrier Captain reported being notified by out going Flight Attendant that a Lithium Ion battery was dropped in a row and not recovered. After extensive search the battery was recovered and flight departed safely.

Narrative

Prior to boarding the aircraft out going FA's (Flight Attendants) advised us of a lithium battery having been dropped at row 23 seat A. They then left without further information. I immediately called Maintenance to help with finding the battery. Additionally; the cleaners were just finishing and I asked if they had found the battery and to dump the contents of the vacuum to see if it was in it. They said they cleaned row 23 but then quickly left without checking the contents of the vacuum. I began searching one row behind and in front so as to not hit the battery by stepping on it. ZZZ Maintenance arrived and began the search as well. At some point a woman not in uniform arrived. I had noticed the floor was filthy as well as the side wall. I made reference that this was making the search difficult. During the search this unknown woman began pulling seat cushions off seats in row 23. I had already discussed with ZZZ Maintenance and was told pulling the cushions wasn't necessary. Yet here was this person yanking seat cushions off. I asked if she was a Mechanic or a Maintenance Supervisor and received a very terse response that she was a manager. So no she was not and was in the row kicking around the debris and possibly the battery. I asked her to please stop and allow Maintenance to do the search. Along with giving me some time to figure out what to do. Her terse response along with telling me I have no authority and she does was enough for me to remove myself from the situation. I made a call to Operations to have her removed then I returned to my duties. Maintenance had done a quick search then left; there was an open log book entry. Maintenance informed my FO (First Officer) that he didn't find it and if we didn't feel safe then don't take it. At this point having spoken to our Dispatcher we once again asked for the cabin to be cleaned and more eyes from Maintenance to see if it could be located. Before anyone arrived the battery was found on the jet bridge.In bound crew left without providing any additional information. It was apparent they had not informed their Captain. Cabin cleaning needs to do a better job; the floor was filthy and the side walls as well. Having non qualified personnel getting involved just made the situation worse. Her behavior alone with her insistence that I had no authority and she did only delayed what we were doing. Maintenance seemed unconcerned; apparently it's not common knowledge that a ruptured lithium battery can ignite. FA's need to understand that informing the Captain and getting good information to us is critical. Non qualified personnel need to stay in their area not get in the way of flight deck crews responsibly. An extra set of eyes is one thing but this person's behavior and actions only hindered our effort. Cabin cleaning needs to be done; not just run the vacuum up the isle then run off. Better understanding of what a lithium battery can do.