Can I update my channel state with another node signing offline in person? I create a channel with someone I know in person.  We both lose internet connection.

Can we transact in person with one another signing each others transactions and maintaining our channel states that way?
Yes that is possible. The channel updates are following BOLT 02 (peer protocol) and parts of BOLT 03 (transactions) and BOLT 05 (on chain). All of these things are solely negotiated between you and your peer / channel partner and could happen in an offline world or even on a sheet of paper. (not saying this would be particularly useful) I am also neglecting the regular protocol mates here that are send via the transport layer and noise protocol framework ( BOLT 08 and 01)

Generally as long as you don't have a (force) close the exact channel state is a private matter between you and your peer. This is a design principle within lightning. 

Keep in mind that what you asked and my answer is only of theoretical nature. I don't think it would be very practical to actually do that in a way so that it is compatible with actual software state.