2 of 3 added 813 characters in body

05AB1E, 7 bytes

A modified version of Sok's answer turned out to be one byte shorter than my initial approach.

êŒAÃéθg

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05AB1E, 8 bytes

êÇ¥ΘγOà>

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How it works

I like challenges like this one which lead to a variety of different approaches.

êŒAÃéθg  | Full program.
ê        | Push a sorted and without duplicates version of the input.
 Π      | Generate all substrings (unfortunately not already sorted by length).
  AÃ     | Keep those that also occur in the lowercase alphabet.
    é    | Sort by length.
     θg  | Take the length of the last one.
---------+-------------------------------------------------------------------------------
êÇ¥ΘγOà> | Full program.
ê        | Push a sorted and without duplicates version of the input.
 Ç       | Cast to ASCII values.
  ¥      | Take the deltas.
   Θ     | For each delta, check if it equals 1 (05AB1E truthified, as 1 is the only
         | truthy "boolean" value in this language).
    γ    | Split into chunks of equal adjacent elements.
     O   | Sum each part.
      à> | Extract the maximum and increment it. This results in the length of the
         | longest run of 1s in the  result of γ.