I did some research because there is a somewhat similar question on sister site and the answers don't seem to agree whether it is safe or not at all, it feels a little bit off so I looked myself a bit and here are my insights.
At first - yes, as others' answers and comments already pointed out permethrin turns out to be widely recognized for being highly toxic to cats. It is explicitly remarked as one of it's properties on independent websites like Wikipedia, etc. but also somewhat implicitly suggested for example on a website of a brand using it in its product - in the context of animals, cats aren't mentioned at all but it's just praised for being "safe for dogs" instead. But digging for information a little bit deeper into the smaller print text reveals that it is in fact toxic to cats, but supposedly safe once it dries. However I'm not really convinced.
Permethrin is what is known as a residual insecticide, which means it remains active long after the application. In a paper I found permethrin's residual efficacy on the plant leaves is examined and it turned out that after a week of application it still killed 90% of mosquitoes. One brand uses this insecticide as an active ingredient in a product meant for spraying clothes, surfaces, etc. and claims that it stays active up to 6 weeks. Actually it is a desirable property in this context and I think it should stay on the balcony undisrupted or else the insect infestation could manifest itself quickly again, but that's not really on-topic for this question.
I have also checked for permethrin's physical properties on national pesticide information center and on Wikipedia and it turns out it's vapor pressure is really low (2.15 x 10-8 mm Hg) which basically means it's evaporating remarkably slowly and it further confirms that it is capable of sticking around for a long time after an application.
Permethrin is also a solid at room temperature (its melting point is 34 °C) and in that context it could be deceiving but it's not like it's completely safe once it dries after an application - it's just the organic solvent that is quickly evaporating, permethrin itself is still there, just (in contrast to a liquid solution) it is now immobile and somewhat stuck to the surface. And yes, it's definitely safer once it dries because it would be harder for a cat to lick it off and ingest comparable amounts of it, but that's still concerning because, I am going to stress that again, permethrin is highly toxic to cats.
As other answer already pointed out, permethrin is hydrophobic and thus it is practically insoluble in water. In the context of potential poisoning it's actually good, because it means that it's solubility in saliva is negligible as well which decreases a risk of ingesting, but doesn't eliminate it. Permethrin surface layer could still be scratched and it's particles physically removed and attached during licking, especially considering the sharp papillae on cat's tongue - which could result in ingesting and poisoning.
As for removing that insecticide, the evaporation rate is extremely slow but it is also slowly decomposing in contact with oxygen in the air and due to ultraviolet radiation in sunlight. In case you wanted to speed up this and do it manually: removing it with water wouldn't work due to it's insolubility, but it would dissolve in an non-polar organic solvent. I'd guess to use something relatively non-toxic like acetone for that purpose, but it would be the best if you asked in the hardware store where you got your spray insecticide how to safely remove it.
In conclusion, if I were you I would bar my cat from entering the balcony for a fair amount of time from now on, as going there it's not an essential need. I wouldn't value marketing claims of safety over the health and life of my pets either. Please consider playing it safe for your cat and don't risk it's well-being and life if it's not needed. Also you could ask your vet about this insecticide because vet's advice would be much more reliable and trustworthy than marketing claims.