"There is not only one but many" is the right option.
Here are some uses from professional publications.
Since, in general, there is not only one but many level dependent
capacities compatible with the preference information provided
by the Decision Maker, we propose to take into account all of
them by using the Robust Ordinal Regression (ROR) and the
Stochastic Multicriteria Acceptability Analysis (SMAA).
— ScienceDirect (with added emphasis)
Ms Ghilaine: There is not only one but many difficulties, and they
are mutually reinforcing.
— Trial International (with added emphasis)
The reason is that, in such cases, the verb must agree with the nearest noun.