Can a digital money system exist, that's secure, anonymous and allows offline transactions?
Depends on what you mean by secure. If you merely want the ability to detect and then presumably punish double spending, you can do that in a way that is secure and anonymous: double spending reveals enough information to provably identify the user. Since honest users don't double spend, they are still anonymous. This is used in Camenisch et al.'s Compact E-cash paper
These systems use techniques related to blind signatures and and similar to anonymous credentials ---e.g. signatures over committed values and zero knowledge proofs about those signatures --- which make it impossible to link the issue of a unit of currency to its spending or the spending of another unit of currency unless it is double spent. As a result, unlike bitcoin, they really are completely anonymous if the cryptographic assumptions hold.
If you mean prevent double spending. No, it's impossible without trusted hardware*. You run up against, among other things, the cap theorem. What stops me from making a clone of my data and executing the same protocol on other ends of the world ( or the galaxy)?
*
As others have pointed out, it might be possible in the quantum setting. Certainly you can make non-cloneable quantum &quot;objects&quot;, but its unclear how you would transfer them.  (update) in fact you can potentially do it with quantum knots3