present progressive

[Julie17]

In a grammar exercise there were the following sentences:

'Can you drive?' - 'I am learning. (A) My father is teaching me.'

Would version B also be correct? (B) '... My father teaches me.'

[English Freak]

In a grammar exercise there were the following sentences:

'Can you drive?' - 'I am learning. (A) My father is teaching me.'

Would version B also be correct? (B) '... My father teaches me.'

We can use present progressive to talk about an action happening around now, and NOT exactly at the time of speaking (that is another use).
Example: Walking in the park, Tom sees Jane after a long time in the park ...
Tom: How are you Jane? Long time, no see. Where have you been?
Jane: Actually I was on a long trip. How about you? Where are you working?
Tom: I'm running my own business.

Here, Tom is in the park, and sure he is not working then! He says "I'm running my own business." and means "these days I'm busy managing my own business."
In your example, I'd go for (A). It sounds better.

[Raymott]

In a grammar exercise there were the following sentences:

'Can you drive?' - 'I am learning. (A) My father is teaching me.'

Would version B also be correct? (B) '... My father teaches me.'

Yes, B is also correct. But it would be more common to stay in present progressive here, I think.