Is all of mathematics based on logic?
Generally speaking, mathematical logic investigates the nature and limitations of logical systems. For example, consider the question: "can every true statement be proved?" This is a problem about logic. When we study geometry, algebra and other branches of mathematics we are interested in using logical thinking, but the purpose is to understand something about shapes or symmetries.
Are there known results in mathematics that are not based on logic?
Well, there's always Ramanujan's approximation of $\pi$ as $\sqrt[4]{\dfrac{2143}{22}}$ , based on a dream which he had one night about a certain Hindu goddess that his family worshipped. :-)
Are there known results in mathematics that are not based on logic?
There are many results in mathematics which are not based exclusively on (first order) logic. A large portion of mathematics is based on set theory, especially ZFC. It would probably be possible to achieve the same results with second order logic (or higher order logic/type theory) with Henkin semantics, and impredicative comprehension axioms emulating the ZFC axioms. However, this is not the current practice how second order logic is used.
Instead of using impredicative comprehension axioms emulating the ZFC axioms, one normally uses much weaker comprehension axioms when using second order logic. The reason is that the people using second order logic instead of set theory do it in the context of reverse mathematics. Here the goal is not just to prove a result based on certain axioms, but also to show that these axioms are really required for proving the result, given a certain base theory like primitive recursive arithmetic.
Given that the asker doesn't understand "mathematical logic", it might seem strange to give an answer based on the differences between first order logic, ZFC set theory and second order logic with Henkin semantics. But maybe this answer helps the asker to realize what he doesn't understand about mathematical logic that prevents him from seeing that the statement "all mathematics are based on logic" might be wrong, if interpreted as "all mathematics are based exclusively on logic".