It is possible to store a digestion of cells with lysis buffer several days at -20 degrees and then proceed with the phenol chloroform isoamyl purification?
Should be fine, but if I were you, I'd spare a few more minutes and carry on until ethanol precipitation (15 min of work at worst). Here's the protocol: https://www.thermofisher.com/pl/en/home/references/protocols/nucleic-acid-purification-and-analysis/dna-extraction-protocols/phenol-chloroform-extraction.html
You can store it in -20 degrees for several days (extend the overnight incubation, that is the second step of ethanol precipitation) and you'll have the certainty that it will stay ok. In fact, freezing in most buffers, whether lysis buffers or extraction buffers shouldn't significantly hurt any nucleic acids. From my personal experience: I've frozen plant material in extraction buffer for RNA isolation (TRI-zol method) in -80 for a few days and got very high concentrations in most samples in the end. I was told freezing in -20 would impact quality but RNA is generally less stable than DNA.
If you had little to none material, you're better of doing the precipitation procedure and then freezing it. If those are minuscule amounts and/or you work with something sensitive like forensic evidence, following through with the procedure will be the most reasonable solution.