MBA short freeze on wake-up

My Macbook Air (13'' mid 2011, 4 GB RAM, 1,7Ghz i5, macOS 10.11.6) is freezing for around 10 seconds when I open it - i.e. wake it up from standby. I see the login screen, it asks me for the password, but I can type and type, no reaction, no blinking cursor. After 5-10 seconds cursor comes on, it catches the last 2-5 characters I typed.

Anybody else experiencing this problem? What can be done?

I am aware that it could be due to its age and processor speed, but it did not freeze that way for many years. I will upgrade as soon as the new MB Pro 13'' is finally (finally, finally) launched. But I will pass the Air onto my wife - so it would be great to solve that issue.

Thanks