Learn Turkish: Ben yalan söylemiyorum.

The verb root is söyle- (to say or tell). To make it negative in the present continuous tense, we add a negative suffix before -yor. Because the root ends in 'e' (söyle-), the negative suffix narrows to -mi to fit vowel harmony. Then we add -yor and the personal ending -um, giving us söylemiyorum.

Why are there two words (yalan söylemiyorum) for "lying"?

In English, "to lie" is a single verb. In Turkish, the action is expressed as a phrase: yalan söylemek, which literally translates to "to tell a lie". yalan means "lie" and söylemek means "to tell". So you are literally saying "I am not telling a lie."

Is the word Ben (I) required here?

No, it is completely optional. The -um ending on söylemiyorum already shows that the subject is "I". Including Ben just adds a bit of emphasis, as if you are saying "I, personally, am not lying."

Why is the personal suffix -um and not -im, given the 'e' and 'i' vowels earlier in the word?