There are two kinds of morphology: inflectional and derivational. The first is like English –ed or –s, or case endings in languages like German or Latin. Adding an inflectional morpheme doesn’t make a new word, but it changes the grammatical function.
Derivational morphology, on the other hand, does make a new word. It might be a different part of speech, like sing > sing-er (verb to noun – person that verbs). Or it might change the meaning in some other way, like call > re-call.
Linguists who study this believe that Old Chinese did not have inflectional morphology, but it did have derivational morphology. Here is just one example from Laurent Sagart’s book The Roots of Old Chinese:
*s- prefix changing noun to verb:
帚 zhou3 (*tuʔ) a broom →
掃 sao3 (*s-tuʔ) to sweep
There were suffixes, too, such as *-s, but when the tones developed many of these disappeared and were preserved only in a tonal change. This can explain some tonal differences like the two pronunciations of 好 (tone 3 vs. tone 4 – the tone 4 one originally had an –s suffix.)
This is really advanced stuff, and I’m not sure if there’s a lot of agreement among the handful of linguists that study it as to exactly what prefixes and suffixes existed in Old Chinese, and what uses they had. If you’re curious, there’s an exciting new book that just came out on this topic. It’s called Old Chinese: A New Reconstruction by William Baxter and Laurent Sagart.