Cisco C6500 outside static nat fails

Hi!

It's just a very simple outside NAT which seems to not work how I would expect it to...

The goal of the rule is to help me with a network transition from one company to another and I need to NAT their DNS-IP to our DNS-IP (plenty 'o IoT devices...). (example IPs)

Our DNS-IP: 10.1.1.1

Their DNS-IP: 192.168.1.1

ip nat outside source static 10.1.1.1 192.168.1.1 

The configured "outside" interface is the new network I need to integrate and where the traffic comes from, the "inside" interface is our existing network and where the traffic goes to.

Now if I try to access 192.168.1.1 everything is fine and the name resolution works... If I try to access our DNS-IP 10.1.1.1 (which should work also) it gets translated to "192.168.1.1" and returns "network unavailable" on icmp.

Do you have tips on how to prevent the translation of our DNS to the other "external"?

Thanks!