F? Pah! Child's play.

Bb - now there's a key for fingerpicking in. A lot of players think that jazzy/ragtimey tunes in keys like F, Bb and EB are bound yo be difficult. But actually, for tunes in that style, most of the passing chords are quite straightforward G, C, D chords, etc. and you only get to hit the tonic chords here and there. Here's an example - composed to demonstrate the principle:

Bb Breakdown