The demand private coded caching problem in a multi-access network with $K$ users and $K$ caches, where each user has access to $L$ neighbouring caches in a cyclic wrap-around manner, is studied. The additional constraint imposed is that one user should not get any information regarding the demands of the remaining users. A lifting construction of demand private multi-access coded caching scheme from conventional, non-private multi-access scheme is introduced. The demand-privacy for a user is ensured by placing some additional \textit{keys} in a set of caches called the \textit{private set} of that user. For a given $K$ and $L$, a technique is also devised to find the private sets of the users.