I've never tried silk, but you can get rayon/bamboo bed sheets that are also silky smooth and feel amazing. They're cool against the skin and I get mine from hoteldirect on amazon. These ones: https://www.amazon.com/Hotel-Sheets-Direct-Bamboo-Piece/dp/B01N9L2O9P/ref=sr_1_3?ie=UTF8&qid=1544245654&sr=8-3&keywords=bamboo+bed+sheets
Search for "bamboo bedsheets" on amazon and it's the first link.
Sheet's that pill up are a crime against all humanity!
We tried bamboo sheets from listing below and they sleep cool, no pilling, silky smooth and we love them. There on all of our beds, we wont buy any others.
Like all reviews everything is objective or subjective depending on which lens your looking through.
HotelSheetsDirect Bed Sheets Set - 1600 Thread Count
https://www.amazon.com/Hotel-Sheets-Direct-Bamboo-Piece/dp/B01N9L2O9P
Recently got these bamboo sheets because they were highly recommended here on reddit(forget where exactly, I just searched for sheets).
For $45, they feel much more luxurious than they should. So. Soft.
Yes, bamboo is the answer, but NOT the cheap stuff that's just Rayon with a greenwashed label. We have this set and they are still amazing 3 years in
I'm in love with these bamboo sheets: https://www.amazon.com/gp/product/B01N9L2O9P/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1. They're silky feeling with a beautiful drape. By your description, I think you may be wanting something with a fuzzy texture, but then you also mention satin which is a totally different texture, so I'm not sure. These are in the satin-like camp... very smooth.
Fight for those sheets! You know amazon has a fantastic array of prices for these; I got my first few sets from the same brand for $46 (amazing deal really) and then when the white ones were briefly out of stock I went with a set at $85 (my bad about the price discrepancy, I remembered incorrectly). Both are really awesome and I always look forward to feeling them on my skin!