Fine liners that don't wear out quickly?
The smaller the tip, the faster it's going to wear out; 0.1 and 0.05 especially tend to wear out and break noticeably faster than wider tips, even with a light touch. There are only two solutions I personally have found for this issue:

1. Buy a cheaper pen and replace it when the tip breaks. There's some affordable brands of fineliners that do come in 0.1 (and sometimes smaller) that are easy enough to purchase in multiples and toss when the tip becomes too damaged to use, and you'll have more choices in brand. I believe some do not have sizes below 0.1 available, but you can also mix and match between brands to find the sizes you want.

2. Buy a more expensive pen and replace only the nib when the tip breaks. I currently use this variety, which is more expensive than the average disposable pen. The nibs are interchangeable and can be replaced fairly easily. Some brands available in small sizes (down to 0.03), but depending on where you live may be harder to find. Additionally, depending on what's available around you, the cost per nib may be equivalent to the cost per pen on a cheaper brand, or the cost for the pen itself far more than reasonable in exchange for disposing of cheaper pens.

You'll have to make the decision on what's the best option for you. I personally chose replaceable nibs due to frequently working away from home and looking for an option that used less space than carrying extra full pens; if you work close to home or a consistent location where you can store more equipment, you may wish to choose whichever option is most cost effective for you. You may also have preferences on the feel of the pen in your hand (my reusable liners have metal bodies, my single use liners are plastic, giving them a different weight and feel), or on the amount of work needed to maintain the pens (it's much easier to toss a pen and grab a new one).

You may also wish to consider a metal nib over plastic (these pens fall into the "replaceable nib" category), but they are less effective when held at an angle and require considerably more maintenance than plastic nibs; again, this is a case where you'll have to personally weigh the benefit (longer life) versus the cost (higher price and frequent cleaning).

For a concise guide to 14 different brands of pens covering a range of prices, nib sizes,  nib materials, and replacement options, JetPens has a review of liners with a quick comparison table at the end, as well as more detailed information on how to maintain the different types of pen available. Ultimately, you'll have to make the decision on what pen might work best for you, based on your own budget, ability/desire to perform any required maintenance, availability in your location, and less tangible preferences (i.e. how the pen feels in your hand or handles in use). I would recommend taking the information from the reviews, then going to a local seller and handling the pens in person (testing when testers are available) to help find the right pen for you.