I'm afraid I'm a bit confused: by definition, a manifold locally looks like an Euclidean space (https://en.wikipedia.org/wiki/Manifold ). On the other hand, if you are interested in an example of a space which is locally Euclidean but not second countable, please see the link below:

http://math.stackexchange.com/questions/25845/does-there-exist-a-connected-locally-euclidean-space-that-is-not-second-countabl

Dear maedeh,

Either you have to modify your first condition not to be a manifold and search for a space that is 2nd countable and Euclidean locally or change the condition of being a manifold in the first place and find what you need. The reason as dear Artur indicated is that a manifold by definition is topological space which is locally Euclidean at every point of it.