You can find a proof of this in the paper

MR1885215 (2002m:57019) Leininger, Christopher J.(1-TX); Reid, Alan W.(1-TX) The co-rank conjecture for 3-manifold groups. (English summary) Algebr. Geom. Topol. 2 (2002), 37–50 (electronic).

It works for any surface of genus at least $2$.

It was originally proven by Casson and by Johnson-Johannson, though they did not publish their proofs (they don't mention the pseudo-Anosov condition, only that the mapping classes do not extend over any handlebody; however, I'm pretty sure that you can get a pseudo-Anosov mapping class by following their proofs). I have a photocopy of the preprint of Johnson-Johannson; if you want it, let me know and I can scan it.

You might also be interested in the paper "Relative Weight Filtrations on Completions of Mapping Class Groups" by Hain (available here) and the thesis of Jamie Jorgensen (available here). They prove that there exist mapping classes that don't extend to any handlebody arbitrarly deep in the "Johnson filtration" of the mapping class group (so these are very algebraically complicated). I'm pretty sure you can follow their ideas to get ones that are pseudo-Anoson.

You can find a proof of this in the paper

MR1885215 (2002m:57019) Leininger, Christopher J.(1-TX); Reid, Alan W.(1-TX) The co-rank conjecture for 3-manifold groups. (English summary) Algebr. Geom. Topol. 2 (2002), 37–50 (electronic).

It works for any surface of genus at least $2$.

It was originally proven by Casson and by Johnson-Johannson, though they did not publish their proofs (they don't mention the pseudo-Anosov condition, only that the mapping classes do not extend over any handlebody; however, I'm pretty sure that you can get a pseudo-Anosov mapping class by following their proofs). I have a photocopy of the preprint of Johnson-Johannson; if you want it, let me know and I can scan it.

You might also be interested in the paper "Relative Weight Filtrations on Completions of Mapping Class Groups" by Hain (available here) and the thesis of Jamie Jorgensen (available here). They prove that there exist mapping classes that don't extend to any handlebody arbitrarly deep in the "Johnson filtration" of the mapping class group (so these are very algebraically complicated). I'm pretty sure you can follow their ideas to get ones that are pseudo-Anoson.