Wolfram Language (Mathematica), 15 bytes

2#-#~Erosion~1&

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Or (39 bytes):

MorphologicalPerimeter[#,Padding->1]+#&

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What else would we expect from Mathematica? Characters used are {Astan -> 0, Blandia -> 1, Trench -> 2}.

K (ngn/k), 23 bytes

{x+x&2{++/'3'0,x,0}/~x}

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uses 0 1 2 for "AB+"

{ } function with argument x

~ logical not

2{ }/ twice do

• 0,x,0 surround with 0-s (top and bottom of the matrix)




• 3' triples of consecutive rows




• +/' sum each




• + transpose

x& logical and of x with

x+ add x to

JavaScript (ES7),  84  82 bytes

Saved 2 bytes thanks to @Shaggy

Takes input as a matrix of integers, with $3$ for Astan and $0$ for Blandia. Returns a matrix with the additional value $1$ for the trench.

a=>(g=x=>a.map(t=(r,Y)=>r.map((v,X)=>1/x?t|=(x-X)**2+(y-Y)**2<v:v||g(X,y=Y)|t)))()

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Octave, 37 31 bytes

This function performs a morphological dilation on the Astan (1-b) part of the "image" using conv2 imdilate, and then uses some arithmetic to make all three areas different symbols.

@(b)imdilate(1-b,ones(3))-1+2*b

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APL (Dyalog Unicode), 11 bytes^SBCS

⊢⌈{2∊⍵}⌺3 3

this is based on @dzaima's 12-byte solution in chat. credit to @Adám himself for thinking of using ∊ in the dfn, and to @H.PWiz for reminding us to use the same encoding for input and output

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represents 'AB+' as 2 0 1 respectively

{ }⌺3 3 apply a function to each overlapping 3×3 region of the input, including regions extending 1 unit outside the matrix, padded with 0s

2∊⍵ is a 2 present in the argument? return a 0/1 boolean

⊢⌈ per-element max of that and the original matrix

MATL, 11 8 bytes

Inspired by @flawr's Octave answer and @lirtosiast's Mathematica answer.

EG9&3ZI-

The input is a matrix with Astan represented by 0 and Blandia by 1. Trench is represented in the output by 2.

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How it works

E       % Implicit input. Multiply by 2, element-wise

G       % Push input again

9       % Push 9

&3ZI    % Erode with neighbourhood of 9 elements (that is, 3×3) 

-       % Subtract, element-wise. Implicit display

Charcoal, 20 bytes

≔⪫θ⸿θＰθＦθ⎇∧№ＫＭA⁼Bι+ι

Try it online! Link is to verbose version of code. Explanation:

≔⪫θ⸿θ

Join the input array with carriage returns rather than the usual newlines. This is needed so that the characters can be printed individually.

Ｐθ

Print the input string without moving the cursor.

Ｆθ

Loop over each character of the input string.

⎇∧№ＫＭA⁼Bι

If the Moore neighbourhood contains an A, and the current character is a B...

+

... then overwrite the B with a +...

ι

... otherwise print the current character (or move to the next line if the current character is a carriage return).

C# (Visual C# Interactive Compiler), 392 bytes

I'm kinda ashamed to post this.

a=>{a=a.Select(b=>b.Prepend(0).Append(0)).Prepend(Enumerable.Repeat(0,a.Count+2)).Append(Enumerable.Repeat(0,a.Count+2)).Select(x=>x.ToList()).ToList();return a.Select((b,c)=>c<1|c==b.Count()-1?b:b.Select((d,e)=>d>1|e<1|e>b.Count-2?d:a[c-1].Skip(e-1).Take(3).Concat(a[c+1].Skip(e-1).Take(3)).Contains(2)|b[e-1]>1|b[e+1]>1?6:d)).Skip(1).Take(a.Count-2).Select(c=>c.Skip(1).Take(c.Count()-2));}

What it does is pad the whole matrix with 0s to avoid index out of bounds exception. Then it checks for each Blandia cell if an Astanian one is close by. Uses 2 for Astan, 1 for Blandia, 6 for trench.

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Retina 0.8.2, 92 80 bytes

(?<=¶(.)*)B(?=.*¶(?<-1>.)*(?(1)_)A|(?<=¶(?(1)_)(?<-1>.)*A.*¶.*))

a

iT`Ba`+`.?a.?

Try it online! Loosely based on my answer to Will I make it out in time? Explanation: Any Bs immediately above or below As are turned into as. This then reduces the problem to checking Bs to the left or right of As or as. The as themselves also need to get turned into +s of course, but fortunately the i flag to T only affects the regex match, not the actual transliteration, so the As remain unaffected.

Ruby, 102 bytes

->a{a+=?.*s=a.size

(s*9).times{|i|a[j=i/9]>?A&&a[j-1+i%3+~a.index($/)*(i/3%3-1)]==?A&&a[j]=?+}

a[0,s]}

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input/output as a newline separated string