Quicklisp is excellent for finding & installing Lisp libraries. I don't know that nano libraries are a good thing — the world probably needs a decent strings package more than it needs left-pad, right-pad, count-chars &c.
>do any of the lisps have a package system like npm with tiny, nano libraries?
In the Common Lisp world there is Quicklisp which works perfectly, however "nano libraries" a la NPM are often shunned in that community (as well as in many others).
Often a Common Lisp library will only require 2 or 3 dependencies and each one might require 0, 1, or 2 dependencies.
Part of this is due to the comprehensive "standard library" built into the language specification itself, part is also due to maturity -- some third party libs have become de-facto standards.
ASDF appears to have been deprecated in favor of Quicklisp, or rather, Quicklisp appears to be the next generation of ASDF, since I saw asdf functions inside of Quicklisp.
> ASDF appears to have been deprecated in favor of Quicklisp
No, ASDF and Quicklisp do different things. ASDF handles compiling and loading systems, whereas Quicklisp handles downloading systems and their dependencies. ASDF is kind of like make, and Quicklisp is kind of like a package manager.