Verbal contract with commercial landlord or just not valid?

I have a retail storefront with a second floor above that is only accessible through the shops on either side of me. When first shown the space, I was informed that the second floor space was mine to utilize for buildout, storage, etc, by my company, and would not be charged rent on that portion. I have a witness (my agent), but it is not in written into the contract. However, now the new neighbor insists that it is written into their contract that they can use the space for their shop. They have a childrens store ans want to use it as a walkway to get to another part of the building for birthday parties. I do not want screaming children running across basically my ceiling (I can hear everything up there).
I asked the landlord's liason many times about the discrepancy and she assured me that the space was mine not theirs, don't worry. Theyare ignoring my requests for a resolution. I cannot afford a lawyer. Do I have a leg to stand on? Is this considered a verbal contract?

You don't say where you live, but in the US you would be hobbling on only one leg.

You have a written lease for the downstairs space. If that lease states that it is the entire agreement between you and the landlord, you will have a very tough time making a case for the upstairs space. The general rule is that the court will not even consider evidence of a verbal agreement when it has a complete written contract to look at. Only what is in the written contract will matter.

Most US state have a 'Statute of Frauds' that requires certain types of contracts must be in writing to be legally enforceable. Most such laws include any contract involving real estate, any contract whose value exceeds a stated amount, and any contract lasting more than a year. I think your verbal agreement would fail on all of these criteria. Also, if ANY of your contract is in writing, you DON'T have any enforceable oral contract in ANY state. A written contract overrides a prior oral contract and can only be changed by a written contract.