Implementing Python's split() function using purely recursion

I'm attempting to implement Python's split() function using recursion with no additional parameters and no loops.

For a given input string, this is the desired output

mySplit('hello,there,world', ',')
=> ['hello', 'there', 'world']

Here is my current attempt, but it really only removes the delimiter and places the string in a list, but I cannot figure out how to append items to the list!

def mySplit(string, delim):
    if len(string) == 1:
        return [string]

    if string[0] == delim:
        return [mySplit(string[1:], delim)[0]]

    return [string[0] + mySplit(string[1:], delim)[0]]

This code results in ['hellothereworld']

I'd write something like:

def my_split(s, delim):
    for i, c in enumerate(s):
        if c == delim:
            return [s[:i]] + my_split(s[i + 1 :], delim)

    return [s]

EDIT: Oops, skipped over a crucial part of your question. I think this works.

def my_split(s, delim, i=0):
    if i == len(s):
        return [s]

    elif s[i] == delim:
        return [s[:i]] + my_split(s[i + 1 :], delim)

    return my_split(s, delim, i + 1)

EDIT 2: It's a tricky one for sure. Really interesting problem. Hopefully I don't hit any more constraints with this one:

def my_split(s, delim):
    if not s:
        return [""]

    elif s[0] == delim:
        a = my_split(s[1:], delim)
        return "", *a

    b, *rest = my_split(s[1:], delim)
    return [s[0] + b] + rest


assert my_split("hello,there,world", ",") == ["hello", "there", "world"]
assert my_split("hello world!", ",") == ["hello world!"]
assert my_split("hello world!", " ") == ["hello", "world!"]