# Jelly, <s>10</s> 9 bytes

    f⁴L
    ŒgÇ€Ṁ

Explanation:

    f⁴L
    f⁴      -Filter by the character argument.
      L     -Return Length of filtered String.

    ŒgÇ€»/
    Œg      -Group string by runs of characters.
      ǀ    -Run above function on each group.
        Ṁ   -Return the largest in the list.

<a href="https://tio.run/##y0rNyan8/z/tUeMWH66jk9IPtz9qWvNwZ8P///8zgFL5CuX5RTkpiv9zAA" title="Jelly – Try It Online">Try it online!</a>