OK, some motivation why I think this could be true:

After blowing up X we obtain X' and a holomorphic map X'->Y. Positive dimensional fibers of X'->X are rationally chain connected, i.e., rational curves connect any two points. On the other hand, Y is not covered by rational curves by assumption.

Let p be a general point in Y and denote by F_p the fiber of X'->Y over p. The above remarks should imply: if F_p meets a positive dimensional fiber of X'->X, then it already contains this fiber (p general!). This should mean X->Y is a fibration over some non-empty open subset of Y. It is either false or well known, I don't know.

By 'regular over U' I meant f is regular everywhere over U. Sometimes this is called 'almost holomorphic'. For example, a standard projection IP_2->IP_1 is not almost holomorphic.

OK, some motivation: After blowing up X we obtain X' and a holomorphic map X'->Y. Positive dimensional fibers of X'->X are rationally chain connected, i.e., rational curves connect any two points. On the other hand, Y is not covered by rational curves by assumption.

Let p be a general point in Y and denote by F_p the fiber of X'->Y over p. The above remarks should imply: if F_p meets a positive dimensional fiber of X'->X, then it already contains this fiber (p general!). This should mean X->Y is a fibration over some non-empty open subset of Y.

It is either false or well known, I don't know.