Do all decidable problems lie in the class NP? All decision problems (i.e.language membership problems), which are verifiable in polynomial time by a deterministic Turing machine are called NP problems. Further, these problems can be solved by a non-deterministic Turing machine in a polynomial time and in exponential time by a deterministic Turing machine.

Do we have a decision problem that is not verifiable by a deterministic Turing machine in polynomial time but decidable?
No to the question in your title; yes to the question in the body of your post.

By the non-deterministic time hierarchy theorem, every decidable NEXP-hard decision problem is "a decision problem that is not verifiable by a deterministic Turing machine in polynomial time but decidable".