Occasionally, when I put my foot down in second gear, the handbrake light flashes for a split second and I hear 3 beeps as if its warning me that something's knackered.

This is becoming more frequent, but only in second gear with a bit of right foot (not ragging it, just brisk acceleration).

There's no drop in performance, no stutter or anything like that, just this beep and handbrake light.

Has anyone had a similar problem, or know what it might be?

I had a similar issue and turned out to be a faulty caliper leaking, the warming was for low brake fluid.

Have you topped up your brake fluid?

Should solve the light flashing as the level of the fluid changes under acceleration.

You'll need to find where its leaking out from though.

Cheers, I'll have to look into it tomorrow.