Pilling is a characteristic of manufactured fibers such as polyester, acrylic, and nylon. Abrasion from normal wear causes the fibers to unravel and the loose ends ball up on the fabric surface. The looser the fibers are spun, the more likely the fabric will pill.
Pilling is commonly found on socks, sweaters, slacks, shirt collars and cuffs, around pockets or on any area subjected to abrasion. Since manufactured fibers are very strong, the pills do not break off completely and can be difficult to remove. In addition, lint can become enmeshed in the balls of fiber which makes the pilling appear more obvious.
Natural fibers like cotton, linen, or wool may pill but the balls of fiber are usually removed during laundering. The lint that results is usually rinsed away at the end of the wash cycle. However, some of the lint from these natural fibers can be picked up by items that attract lint if they are washed or dried in the same load
I recommend getting a pair of Nylon Hosiery and ball it up and wipe every square inch of the surfaces inside the basket. If the washer is causing the pilling the Nylon Hosiery will find the area causing it. The Nylon Hosiery will snag on any area in the basket that could relate to the pilling.
Be sure not to overload the washer
and to sort clothes accordingly. Lint can be trapped in the load if overloaded. Wash smaller loads. Separate lint-givers from lint-takers: Towels and cotton fleece do not mix well with polar fleece or corduroy. Pilling results from rubbing (abrasion) of the fabric during normal wear and use. While pilling cannot be eliminated it can be minimized by proper handling during washing of the fabric/garment. Before laundering, turn the garment inside and out. And, remove the garment from the dryer as soon as it is dry.
To remove any pills on fabric, pull the fabric taut over a curved surface and carefully cut off the pill with scissors or shave the fabric surface with a safety razor. There are also battery operated pill removers, which shave the pills much like an electric razor.
However, it's important to understand that once you remove the pills, they can come back. So you may find that you'll have to remove pills from time to time to keep your garment looking fresh and new.