Is it possible to use LIKE and IN for a WHERE statment?

It's a common misconception that for the construct

b IN (x, y, z)

that (x, y, z) represents a set. It does not.

Rather, it is merely syntactic sugar for

(b = x OR b = y OR b = z)

SQL has but one data structure: the table. If you want to query search text values as a set then put them into a table. Then you can JOIN your search text table to your Places table using LIKE in the JOIN condition e.g.

WITH Places (Name)
     AS
     (
      SELECT Name
        FROM (
              VALUES ('Ballymeade Country Club'), 
                     ('Ballymena Candles'), 
                     ('Bangers & Mash Cafe'), 
                     ('Bangebis')
             ) AS Places (Name)
     ), 
     SearchText (search_text)
     AS
     (
      SELECT search_text
        FROM (
              VALUES ('Ballymena'), 
                     ('Banger')
             ) AS SearchText (search_text)
     )
SELECT * 
  FROM Places AS P1
       LEFT OUTER JOIN SearchText AS S1
          ON P1.Name LIKE S1.search_text + '%';

No, but you can use OR instead:

WHERE (dbo.[Places].[Name] LIKE 'Ballymena%' OR
       dbo.[Places].[Name] LIKE 'Banger%')