iPad Mini 1 won't update from iso 9.3.5 to iso 13.5.1, why?

Your very old first generation iPad mini can only be updated to iOS 9.3.5 (WiFi Only models) or iOS 9.3.6 (WiFi & Cellular models).

Apple ended update support for iPad mini in September 2016. Your iPad mini cannot be updated to iOS 10 or later major versions of iOS, as the internal hardware does not meet the minimum technical requirements for new versions of iOS/iPadOS. This limitation cannot be bypassed - in addition to which, iPad mini is a 32-bit architecture device, whilst all recent versions of iOS/iPadOS are designed for 64-bit hardware platforms.

You may wish to consider that compatible Apps for your iPad are increasingly difficult to find - as third-party developers are increasingly dropping support for older versions of iOS preceding iOS11/12/13. As older versions of Apps are removed from the Apple App Store, you’ll not be able to (re)download or install them. As such, if you want or need to access most current Apps, requiring more recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model.

Even the most basic current iPad7 will massively outperform you current iPad - and is compatible with the vast majority of current Apps.

I hope this information provides helpful clarity as to your available options - and the limitations of your current iPad mini.

A 1st gen iPad Mini cannot be upgraded beyond iOS 9.3.5/9.3.6 for WiFi AND cellular data models.

A 1st gen iPad Mini is a NEARLY 8-year old device, now!

These older iPad models are too old, now, too underpowered and completely incompatible with any of the current and future versions of iOS/iPadOS.

It is what it is, today, in the year 2020!

Apple supports new iPad models for up to 5-6 years with third party app developers supporting these iPads for another, additional 2-3 years after!

So, that is a total of between 7-9 years of iPad/iOS support!

ALL older iPad models, like the iPad 2, 3, 4 and 1st gen iPad Mini have, finally, reached this support limit!

If you need an iPad that can run the latest and greatest versions of iOS, as well as, the most current apps available, then, you need to purchase a new iPad from Apple or a discounted, refurbished iPad from Apple’s Refurbished & Clearance section of Apple’s Website or a new or refurbished iPad from an Apple authorised, third party retail or online reseller OR other trusted sources for new or used Apple products.

These old iPad 2, 3, 4 and 1st gen iPad Mini models are, virtually, 8 and 9-years old, now.

The amount of older, compatible, available apps for these old iPad models has dwindled significantly since the beginning of 2020!

Getting older versions of apps for these old iPad models, initially from Apple’s iOS App Store is getting very difficult and very problematic and this issue is getting, painfully, much worse, now, in 2020!

As of the beginning of 2019, many, MANY third party app developers are all more quickly and completely permanently dropping older app support for all old 7, 8 and 9-year iPads that are running 2-3 year old and earlier iOS versions ( iOS 10 and earlier ).

Third party app developers are removing all of their older, legacy versions of their apps from Apple's App Store servers, now.

Third party app developers no longer want to support these 8 and 9-year old or older iOS devices, any longer!

Older versions of apps are, soon, going to be gone for good, now!

These old iPads are now reaching their complete end of life and support cycles.

So finding older versions apps for older iPads, while still possible, currently, will be a very tough challenge, now, in 2020.

Old iPads that can no longer be upgraded/updated will have NO MORE APPS AVAILABLE for initial download, at all, soon, from the Apple iOS App Store, for such old iOS devices in the very near future.

Older versions of apps for these older iPads will be completely gone, at some point, in the very foreseeable future.

Old 8 and 9-year old iPads will be pretty much obsolete by the end of the 2020 year!

Again, it is what it is, today, in the year 2020!

You will never get any more older versions of popular apps for these old iPads, any longer.

The apps that these old iPads already have on them, and still work, will be the apps these iPads will be stuck with!

Basically, these old iPad models are just good for FaceTime and somewhat limited Web browsing via some other, more up to date third party Web browsers and NOT using that nearly 4-year old iOS Safari Web browser.

So, this is REALLY NOW the time to seriously consider a new or discounted, refurbished iPad from Apple or other Apple authorised resellers.

A lot has happened with iPad and iOS in 7-9 years and in the past 2 to 3-1/2 years since iOS 9 and 10.

As an example and compared to that nearly 8-years old iPad Mini model,

The cheapest, baseline 2019, 10.2 inch screen 7th gen Apple iPads have a 1/2 inch larger retina screen, it is overall lighter, in weight, newer, more powerful and approx. 2-1/2 times faster 2.3 Ghz, A10 Fusion processor ( CPU ), AT LEAST 3-4 times the internal RAM, with the 2019, 7th gen iPad having 3 GBs of internal RAM, iPadOS 13 support, with much better and improved multitasking features and much fuller support for externally connected data drives, Apple Pencil and Apple Smart keyboard cover support and have better audio speakers and better/improved built-in cameras.

In addition, if you search online and look at various, reputable online Apple device resellers, you can usually find this 2019, 7th gen iPad at various discounts, depending on storage size and whether it is a WiFi ONLY model or a WiFi/cellular data model.

This is the harsh reality for these 8 and 9-year old, OR OLDER, iOS devices, now!

Sorry & Best of Luck to You!

Per my original reply, your iPad mini cannot be updated to any later version. It has already reached its terminal/final update. This hard-limitation cannot be bypassed.

If you want or need to access most current Apps, requiring a more recent version of iOS that your current iPad cannot support, you’ll need to consider replacement of your iPad with a newer or current model.