Use named Pattern to extract a part of matched substring:
StringCases[#, str : Shortest[StartOfString ~~ ___] ~~ "!" :> str] & /@ s
{{{"This is small string 2 "}}, {{"There is string 5 "}}, {{"This is String n "}}}
or
StringCases[#, str : Shortest[StartOfString ~~ ___] ~~ "!" :> str] &@Flatten@ s
{{"This is small string 2 "}, {"There is string 5 "}, {"This is String n "}}
As Martin notes in the comments, another approach is to capture Longest sequence of characters from a negated character class. In this case we don't have to use a named pattern which introduce additional overhead, hence this approach should be more efficient. Since string patterns by default are greedy, Longest can be omitted in this case:
StringCases[#, StartOfString ~~ Except["!"] ...] &
Translating StringExpressions into equivalent regular expressions sometimes gives substantial increase in performance: this is what Mathematica always does under the hood, but not always in the optimal way. Here is verbatim semantic translation:
StringCases[#, RegularExpression["^[^!]*"]] &
(read here on how to find out what a regex Mathematica generates from a StringExpression).
And another regex without capturing group and without use of a negated character class:
StringCases[#, RegularExpression["^.*?(?=!)"]] &
Note that this last regex can't be expressed as a combination of usual Wolfram Language patterns because it uses a positive lookahead zero-length assertion which has no equivalent in the world of Wolfram Language symbolic pattern objects.
Performance comparison
Here is a timing comparison of the all suggested solutions (on a very large number of short strings):
$HistoryLength = 0;
s1 = ConstantArray["There is string 5 ! Or is it 2?", 2*10^5];
{First@AbsoluteTiming[# /@ s1], #} & /@ {
StringCases[#, str : Shortest[StartOfString ~~ ___] ~~ "!" :> str] &,
StringCases[#, RegularExpression["^(.*?)!"] -> "$1"] &,
StringCases[#, StartOfString ~~ Except["!"] ...] &,
StringCases[#, RegularExpression["^[^!]*"]] &,
StringCases[#, RegularExpression["^.*?(?=!)"]] &,
StringDelete[#, "!" ~~ ___] &, StringExtract[#, "!" -> 1] &,
StringTake[#, StringPosition[#, "!", 1][[1, 1]] - 1] &,
StringDrop[#, {StringPosition[#, "!", 1][[1, 1]], -1}] &} //
Grid[#, Frame -> All, Alignment -> Left, FrameStyle -> Directive[Thin, LightGray]] &
MaxMemoryUsed[]

102886352
Another comparison suggested by Martin (on a string where it takes a while to find the "!"):
$HistoryLength = 0;
s2 = StringRepeat["There is string 5 ", 10^5] <> "! Or is it 2?";
results = {};
{(AppendTo[results, #2]; #1) & @@ AbsoluteTiming[#@s2], #} & /@ {
StringCases[#, str : Shortest[StartOfString ~~ ___] ~~ "!" :> str][[1]] &,
StringCases[#, RegularExpression["^(.*?)!"] -> "$1"][[1]] &,
StringCases[#, StartOfString ~~ Except["!"] ...][[1]] &,
StringCases[#, RegularExpression["^[^!]*"]][[1]] &,
StringCases[#, RegularExpression["^.*?(?=!)"]][[1]] &,
StringDelete[#, "!" ~~ ___] &,
StringExtract[#, "!" -> 1] &,
StringTake[#, StringPosition[#, "!", 1][[1, 1]] - 1] &,
StringDrop[#, {StringPosition[#, "!", 1][[1, 1]], -1}] &} //
Grid[#, Frame -> All, Alignment -> Left, FrameStyle -> Directive[Thin, LightGray]] &
SameQ @@ results
MaxMemoryUsed[]

True
82707472
The conclusion: Martin's solution via negated character class with greedy quantifier outperforms others in general: RegularExpression["^[^!]*"]. See this dedicated Q&A about why the equivalent string expression StartOfString ~~ Except["!"] ... is two orders of magnitude slower.
StringCases[#, StartOfString ~~ Except["!"] ...] &– Martin Ender Mar 21 '17 at 10:17!, e.g.StringRepeat["There is string 5 ", 100] <> "! Or is it 2?". The negative character class and greedy quantifier are much faster. The problem with the other solution in terms of efficiency is that ungreedy quantifiers require a lot of backtracking. It tries one character, then fails at the lookahead, then it tries two characters and fails at the lookahead, etc etc. The greedy solution just keeps going as long as possible and never has to backtrack. – Martin Ender Mar 21 '17 at 14:21"!". Surprisingly,StringExtractbeats regex in this case. – Alexey Popkov Mar 21 '17 at 14:52