I think at 11 months, a 40 min nap isn't the problem it may be for a 4 month old who still needs a lot of sleep during the day iykwim? Has she always been like this? Despite what the 'experts' tell us, some babies just never extend their naps beyond one sleep cycle despite mum trying and trying to resettle them.

The 'general rule of thumb' for 10-12 months is 2x naps of 1-2 hours. So she's only just cutting it short. Soon I would think she'll be ready to have one big sleep in the middle of the day (this usually happens about 12-15 months from my experience) and many mums who previously had short nappers find that they do better in one block - so she might work her way up to having one big sleep of about 2-2.5 hours.

How is she when she wakes after 40 mins? Is she crying for you? If she is happy enough to talk and play for a while longer, I'd leave her in her bed for around the hour mark, at least she is rested and not being stimulated, it might help.