Can I defer depositing severance pay check to next year?

Sorry ... your employer will add that to the 2019 W2 issued in January ... there is no way to "defer" it by not cashing it until 2020. 

You can cash/deposit the check any time you want. But it does not change the fact that you were paid the money in 2019 and it will be included as a part of your 2019 income. Basically, you will have direct control of that money the day you get the check. Therefore, no matter when you cash it, that money is included in your 2019 income. The only thing you can do to change that, is to tell the employer to keep the severance pay because you don't want it.

Ask them not to pay you until next year?  They could earn interest on it til then.   Or do you have a 401K plan you could put most or all of it in?  That's what I did.  Put it all towards my 401K.

even asking your ex-employer not to issue check until 2020 will not work since you are the one making the decision as to when to get paid